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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 322161e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 322161.e5 | 322161e1 | \([1, 1, 1, -414747, -110245872]\) | \(-53297461115137/4513839183\) | \(-668210196258438687\) | \([2]\) | \(4866048\) | \(2.1653\) | \(\Gamma_0(N)\)-optimal |
| 322161.e4 | 322161e2 | \([1, 1, 1, -6765392, -6775882864]\) | \(231331938231569617/1472026689\) | \(217912779537841521\) | \([2, 2]\) | \(9732096\) | \(2.5118\) | |
| 322161.e3 | 322161e3 | \([1, 1, 1, -6894997, -6502934734]\) | \(244883173420511137/18418027974129\) | \(2726529144777055515681\) | \([2, 2]\) | \(19464192\) | \(2.8584\) | |
| 322161.e1 | 322161e4 | \([1, 1, 1, -108246107, -433522585582]\) | \(947531277805646290177/38367\) | \(5679692953263\) | \([2]\) | \(19464192\) | \(2.8584\) | |
| 322161.e2 | 322161e5 | \([1, 1, 1, -22466112, 33309292098]\) | \(8471112631466271697/1662662681263647\) | \(246133748127987627027183\) | \([2]\) | \(38928384\) | \(3.2050\) | |
| 322161.e6 | 322161e6 | \([1, 1, 1, 6602438, -28843889146]\) | \(215015459663151503/2552757445339983\) | \(-377899717822273290649887\) | \([2]\) | \(38928384\) | \(3.2050\) |
Rank
sage: E.rank()
The elliptic curves in class 322161e have rank \(1\).
Complex multiplication
The elliptic curves in class 322161e do not have complex multiplication.Modular form 322161.2.a.e
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
