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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 4312e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4312.b2 | 4312e1 | \([0, 1, 0, 187556, -33079488]\) | \(24226243449392/29774625727\) | \(-896756465191890688\) | \([2]\) | \(46080\) | \(2.1309\) | \(\Gamma_0(N)\)-optimal |
| 4312.b1 | 4312e2 | \([0, 1, 0, -1116824, -318999584]\) | \(1278763167594532/375974556419\) | \(45294623322254265344\) | \([2]\) | \(92160\) | \(2.4775\) |
Rank
sage: E.rank()
The elliptic curves in class 4312e have rank \(0\).
Complex multiplication
The elliptic curves in class 4312e do not have complex multiplication.Modular form 4312.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
