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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 15225o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 15225.w5 | 15225o1 | \([1, 0, 1, -19601, 1129223]\) | \(-53297461115137/4513839183\) | \(-70528737234375\) | \([2]\) | \(49152\) | \(1.4022\) | \(\Gamma_0(N)\)-optimal |
| 15225.w4 | 15225o2 | \([1, 0, 1, -319726, 69557723]\) | \(231331938231569617/1472026689\) | \(23000417015625\) | \([2, 2]\) | \(98304\) | \(1.7488\) | |
| 15225.w3 | 15225o3 | \([1, 0, 1, -325851, 66752473]\) | \(244883173420511137/18418027974129\) | \(287781687095765625\) | \([2, 2]\) | \(196608\) | \(2.0954\) | |
| 15225.w1 | 15225o4 | \([1, 0, 1, -5115601, 4452987473]\) | \(947531277805646290177/38367\) | \(599484375\) | \([2]\) | \(196608\) | \(2.0954\) | |
| 15225.w2 | 15225o5 | \([1, 0, 1, -1061726, -342394027]\) | \(8471112631466271697/1662662681263647\) | \(25979104394744484375\) | \([2]\) | \(393216\) | \(2.4420\) | |
| 15225.w6 | 15225o6 | \([1, 0, 1, 312024, 296387473]\) | \(215015459663151503/2552757445339983\) | \(-39886835083437234375\) | \([2]\) | \(393216\) | \(2.4420\) |
Rank
sage: E.rank()
The elliptic curves in class 15225o have rank \(1\).
Complex multiplication
The elliptic curves in class 15225o do not have complex multiplication.Modular form 15225.2.a.o
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.
