Sen. Thomas F. Eagleton (D-Mo.) is making few campaign apearances as the trial of his niece, who is charged with trying to extort $220,000 from him, unfolds in federal court here.

Polls indicate that Eagleton, who is seeking a third term in the Nov. 4 election, holds a large lead over his Republican opponent, St. Louis County Executive Gene McNary, who is barn-storming around the state by helicopter. eBut, said one long-time state political observer, "That trial may be the story of the campaign."

The senator's niece, Elizabeth (Libby) (Eagleton Weigand, and her former attorney, Stephen E. Poludniak, are accused of trying to extort the money from Eagleton by threatening to release unspecified damaging material about him.

Prosecutors said Weigand and Poludniak had threatened to release the allegedly damaging material to the St. Louis Globe-Democrat and to a McNary campaign aide unless Eagleton purchased for $220,000 her 6.25 percent interest in a family-owned pipe-fittings company. Weigand was given the stock in the early 1970s by her father under a set of tight restrictions aimed at keeping it in the Eagleton family.

At the time of Weigand's arrest about 2 1/2 months ago, Eagleton said his niece was trying to sell the stock so she could give the proceeds to the Church of Scientology. The senator charged also that the church was involved in the alleged plot. But the church isn't mentioned in the indictment, church officials have denied any wrongdoing, and the topic hasn't arisen in any testimony.

The nature of the alleged threat against Eagleton is something of a mystery. The grand jury didn't specify what it was. Trial jurors have been told only that it was "gossip" of a "tragic nature" that involved Key West, Fla., but the reference to Key West hasn't been explained. A defense attorney told the jury that the material apparently was false and had been intended solely "to get a rise" out of the senator.

Eagleton previously had dismissed the material as "garbage" and "a bunch of nothing." Before the start of the trial, he twice asked the Federal Bureau of Investigation to make it public, but the bureau refused, saying that to do so would be against the law.

It isn't clear whether Eagleton will testify at the trial. Incumbent senators rarely are required to testify under oath in an election year. But when Eagleton disclosed the alleged threat at a news conference the day before the Aug. 5 Missouri primary, he offered to testify against his niece.

As a result, the jurors were questioned extensively about how they would view Eagleton's testimony. But prosecutors haven't disclosed their witness list and won't discuss whether the senator will testify.

Attorney for Weigand and Poludniak will try to portray Eagleton in an unfavorable light. Weigand's attorney, for example, told the jury that "Uncle Tommy" had "shunned" repeated requests by his niece for counsel after her father, Dr. Mark D. Eagleton, suffered a serious stroke.

The attorney, Leonard J. Frankel, also told the jurors that the charges against Weigand grew out of a family dispute involving a "tremendous problem of miscommunication" between the senator and Mark Eagleton's children over alleged irregularities in the family-owned business, the Missouri Pipe Fittings Co. of St. Louis.

Weigand and Poludniak may offer antagonistic defense. Poludniak's attorney didn't make an opening statement, but Frankel told the jury that Weigand wrote out three pages of "gossip" -- later seized by the FBI and expected to be introduced in the trial -- at Poludniak's urging after he told her that to do so didn't violate the law.

The trial already has taken some unusual twists. After rejecting repeated defense requests to move the trial from St. Louis because of pretrial publicity and Eagleton's influence as a senator, Chief U.S. District Judge H. Kenneth Wangelin agreed to a defense request to question all prospective jurors in his chambers about their attitudes toward Eagleton and the Church of Scientology.

The St. Louis Post-Dispatch then sought and obtained an emergency hearing from the 8th U.S. Circuit Court of Appeals here. The appellate court ruled that the closed-door questioning was "inappropriate," and ordered Wangelin to examine in public the jurors who hadn't been questioned.

Eagleton has been in political white-water before. In 1972, Democratic presidential nominee George McGovern chose Eagleton to run for vice president, but Eagleton left the ticket when it was disclosed that he had been hospitalized for psychiatric treatment. Nevertheless, Eagleton has remained popular with Missouri voters.