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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 62197c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
62197.c3 | 62197c1 | \([0, -1, 1, -5603, 162045]\) | \(4096000/37\) | \(175753856917\) | \([]\) | \(44880\) | \(0.98026\) | \(\Gamma_0(N)\)-optimal |
62197.c2 | 62197c2 | \([0, -1, 1, -39223, -2882246]\) | \(1404928000/50653\) | \(240607030119373\) | \([]\) | \(134640\) | \(1.5296\) | |
62197.c1 | 62197c3 | \([0, -1, 1, -3149073, -2149860489]\) | \(727057727488000/37\) | \(175753856917\) | \([]\) | \(403920\) | \(2.0789\) |
Rank
sage: E.rank()
The elliptic curves in class 62197c have rank \(0\).
Complex multiplication
The elliptic curves in class 62197c do not have complex multiplication.Modular form 62197.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.