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SageMath
E = EllipticCurve("br1")
E.isogeny_class()
Elliptic curves in class 111090br
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 111090.bn6 | 111090br1 | \([1, 1, 1, 5279, 208799]\) | \(109902239/188160\) | \(-27854432874240\) | \([2]\) | \(360448\) | \(1.2648\) | \(\Gamma_0(N)\)-optimal |
| 111090.bn5 | 111090br2 | \([1, 1, 1, -37041, 2121663]\) | \(37966934881/8643600\) | \(1279563010160400\) | \([2, 2]\) | \(720896\) | \(1.6114\) | |
| 111090.bn4 | 111090br3 | \([1, 1, 1, -195741, -31586217]\) | \(5602762882081/345888060\) | \(51203846456585340\) | \([2]\) | \(1441792\) | \(1.9579\) | |
| 111090.bn2 | 111090br4 | \([1, 1, 1, -555461, 159099239]\) | \(128031684631201/9922500\) | \(1468886108602500\) | \([2, 2]\) | \(1441792\) | \(1.9579\) | |
| 111090.bn3 | 111090br5 | \([1, 1, 1, -518431, 181272803]\) | \(-104094944089921/35880468750\) | \(-5311597089142968750\) | \([2]\) | \(2883584\) | \(2.3045\) | |
| 111090.bn1 | 111090br6 | \([1, 1, 1, -8887211, 10193858939]\) | \(524388516989299201/3150\) | \(466313050350\) | \([2]\) | \(2883584\) | \(2.3045\) |
Rank
sage: E.rank()
The elliptic curves in class 111090br have rank \(1\).
Complex multiplication
The elliptic curves in class 111090br do not have complex multiplication.Modular form 111090.2.a.br
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
