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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 5712c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5712.d2 | 5712c1 | \([0, -1, 0, 16, 144]\) | \(415292/9639\) | \(-9870336\) | \([2]\) | \(1280\) | \(0.022321\) | \(\Gamma_0(N)\)-optimal |
5712.d1 | 5712c2 | \([0, -1, 0, -344, 2448]\) | \(2204605874/127449\) | \(261015552\) | \([2]\) | \(2560\) | \(0.36890\) |
Rank
sage: E.rank()
The elliptic curves in class 5712c have rank \(2\).
Complex multiplication
The elliptic curves in class 5712c do not have complex multiplication.Modular form 5712.2.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.