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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 24150bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 24150.y5 | 24150bc1 | \([1, 0, 1, 3149, 139598]\) | \(221115865823/664731648\) | \(-10386432000000\) | \([2]\) | \(65536\) | \(1.1810\) | \(\Gamma_0(N)\)-optimal |
| 24150.y4 | 24150bc2 | \([1, 0, 1, -28851, 1611598]\) | \(169967019783457/26337394944\) | \(411521796000000\) | \([2, 2]\) | \(131072\) | \(1.5275\) | |
| 24150.y3 | 24150bc3 | \([1, 0, 1, -126851, -15832402]\) | \(14447092394873377/1439452851984\) | \(22491450812250000\) | \([2, 2]\) | \(262144\) | \(1.8741\) | |
| 24150.y2 | 24150bc4 | \([1, 0, 1, -442851, 113391598]\) | \(614716917569296417/19093020912\) | \(298328451750000\) | \([2]\) | \(262144\) | \(1.8741\) | |
| 24150.y6 | 24150bc5 | \([1, 0, 1, 156649, -76501402]\) | \(27207619911317663/177609314617308\) | \(-2775145540895437500\) | \([2]\) | \(524288\) | \(2.2207\) | |
| 24150.y1 | 24150bc6 | \([1, 0, 1, -1978351, -1071187402]\) | \(54804145548726848737/637608031452\) | \(9962625491437500\) | \([2]\) | \(524288\) | \(2.2207\) |
Rank
sage: E.rank()
The elliptic curves in class 24150bc have rank \(1\).
Complex multiplication
The elliptic curves in class 24150bc do not have complex multiplication.Modular form 24150.2.a.bc
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.
