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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 51714c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51714.j1 | 51714c1 | \([1, -1, 0, -82926, 8880340]\) | \(17923019113/735488\) | \(2587993811290368\) | \([2]\) | \(258048\) | \(1.7230\) | \(\Gamma_0(N)\)-optimal |
51714.j2 | 51714c2 | \([1, -1, 0, 38754, 32656612]\) | \(1829276567/132066064\) | \(-464706638739826704\) | \([2]\) | \(516096\) | \(2.0696\) |
Rank
sage: E.rank()
The elliptic curves in class 51714c have rank \(0\).
Complex multiplication
The elliptic curves in class 51714c do not have complex multiplication.Modular form 51714.2.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 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.