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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 121968.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121968.c1 | 121968dp2 | \([0, 0, 0, -2631387, -1412417270]\) | \(286191179/43218\) | \(304289038470046113792\) | \([2]\) | \(6488064\) | \(2.6544\) | |
121968.c2 | 121968dp1 | \([0, 0, 0, -714747, 210976810]\) | \(5735339/588\) | \(4139986917959811072\) | \([2]\) | \(3244032\) | \(2.3078\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 121968.c have rank \(1\).
Complex multiplication
The elliptic curves in class 121968.c do not have complex multiplication.Modular form 121968.2.a.c
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.