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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 92400f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 92400.a5 | 92400f1 | \([0, -1, 0, -26383, -1456238]\) | \(8124052043776/992578125\) | \(248144531250000\) | \([2]\) | \(393216\) | \(1.4924\) | \(\Gamma_0(N)\)-optimal |
| 92400.a4 | 92400f2 | \([0, -1, 0, -104508, 11512512]\) | \(31558509702736/4035425625\) | \(16141702500000000\) | \([2, 2]\) | \(786432\) | \(1.8389\) | |
| 92400.a6 | 92400f3 | \([0, -1, 0, 157992, 59812512]\) | \(27258770992316/112538412525\) | \(-1800614600400000000\) | \([2]\) | \(1572864\) | \(2.1855\) | |
| 92400.a2 | 92400f4 | \([0, -1, 0, -1617008, 791962512]\) | \(29224056825643684/588305025\) | \(9412880400000000\) | \([2, 2]\) | \(1572864\) | \(2.1855\) | |
| 92400.a3 | 92400f5 | \([0, -1, 0, -1562008, 848282512]\) | \(-13171152353214242/2080257264855\) | \(-66568232475360000000\) | \([2]\) | \(3145728\) | \(2.5321\) | |
| 92400.a1 | 92400f6 | \([0, -1, 0, -25872008, 50660242512]\) | \(59850000883110493442/24255\) | \(776160000000\) | \([2]\) | \(3145728\) | \(2.5321\) |
Rank
sage: E.rank()
The elliptic curves in class 92400f have rank \(1\).
Complex multiplication
The elliptic curves in class 92400f do not have complex multiplication.Modular form 92400.2.a.f
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.
