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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 31218.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31218.f1 | 31218e2 | \([1, 0, 1, -7248024, 7510936090]\) | \(-23769846831649063249/3261823333284\) | \(-5778519006135936324\) | \([]\) | \(1399440\) | \(2.6179\) | |
31218.f2 | 31218e1 | \([1, 0, 1, 19236, -2292710]\) | \(444369620591/1540767744\) | \(-2729564045328384\) | \([]\) | \(199920\) | \(1.6449\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 31218.f have rank \(1\).
Complex multiplication
The elliptic curves in class 31218.f do not have complex multiplication.Modular form 31218.2.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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.