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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 327990.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 327990.r1 | 327990r6 | \([1, 0, 1, -7581633, -8035747934]\) | \(81025909800741361/11088090\) | \(6595454517346890\) | \([2]\) | \(12386304\) | \(2.4477\) | |
| 327990.r2 | 327990r3 | \([1, 0, 1, -710663, 230350238]\) | \(66730743078481/60937500\) | \(36247046123437500\) | \([2]\) | \(6193152\) | \(2.1011\) | |
| 327990.r3 | 327990r4 | \([1, 0, 1, -475183, -124847794]\) | \(19948814692561/231344100\) | \(137608865855756100\) | \([2, 2]\) | \(6193152\) | \(2.1011\) | |
| 327990.r4 | 327990r5 | \([1, 0, 1, -96733, -318160054]\) | \(-168288035761/73415764890\) | \(-43669409085624999690\) | \([2]\) | \(12386304\) | \(2.4477\) | |
| 327990.r5 | 327990r2 | \([1, 0, 1, -54683, 1806806]\) | \(30400540561/15210000\) | \(9047262712410000\) | \([2, 2]\) | \(3096576\) | \(1.7546\) | |
| 327990.r6 | 327990r1 | \([1, 0, 1, 12597, 218998]\) | \(371694959/249600\) | \(-148467900921600\) | \([2]\) | \(1548288\) | \(1.4080\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 327990.r have rank \(0\).
Complex multiplication
The elliptic curves in class 327990.r do not have complex multiplication.Modular form 327990.2.a.r
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
