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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 22253.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22253.e1 | 22253b1 | \([0, -1, 1, -25817, 1605285]\) | \(-78843215872/539\) | \(-13010149691\) | \([]\) | \(30720\) | \(1.1216\) | \(\Gamma_0(N)\)-optimal |
| 22253.e2 | 22253b2 | \([0, -1, 1, -14257, 3034390]\) | \(-13278380032/156590819\) | \(-3779721698379011\) | \([]\) | \(92160\) | \(1.6709\) | |
| 22253.e3 | 22253b3 | \([0, -1, 1, 127353, -78462165]\) | \(9463555063808/115539436859\) | \(-2788841129405255771\) | \([]\) | \(276480\) | \(2.2202\) |
Rank
sage: E.rank()
The elliptic curves in class 22253.e have rank \(2\).
Complex multiplication
The elliptic curves in class 22253.e do not have complex multiplication.Modular form 22253.2.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 LMFDB numbering.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
