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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 12789.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12789.f1 | 12789g2 | \([1, -1, 1, -15665, 751538]\) | \(4956477625/52983\) | \(4544146388943\) | \([2]\) | \(24576\) | \(1.2445\) | |
| 12789.f2 | 12789g1 | \([1, -1, 1, -230, 29180]\) | \(-15625/4263\) | \(-365620973823\) | \([2]\) | \(12288\) | \(0.89795\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12789.f have rank \(1\).
Complex multiplication
The elliptic curves in class 12789.f do not have complex multiplication.Modular form 12789.2.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 LMFDB 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 LMFDB labels.
