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SageMath
E = EllipticCurve("cl1")
E.isogeny_class()
Elliptic curves in class 69360cl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
69360.i2 | 69360cl1 | \([0, -1, 0, -70901, -420099]\) | \(115220905984/66430125\) | \(22725879685632000\) | \([]\) | \(497664\) | \(1.8281\) | \(\Gamma_0(N)\)-optimal |
69360.i1 | 69360cl2 | \([0, -1, 0, -3816341, 2870834205]\) | \(17968412610002944/158203125\) | \(54121608000000000\) | \([]\) | \(1492992\) | \(2.3774\) |
Rank
sage: E.rank()
The elliptic curves in class 69360cl have rank \(1\).
Complex multiplication
The elliptic curves in class 69360cl do not have complex multiplication.Modular form 69360.2.a.cl
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.