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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 13923j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
13923.d1 | 13923j1 | \([1, -1, 1, -410, -2896]\) | \(10431681625/710073\) | \(517643217\) | \([2]\) | \(6144\) | \(0.42059\) | \(\Gamma_0(N)\)-optimal |
13923.d2 | 13923j2 | \([1, -1, 1, 355, -12994]\) | \(6804992375/102626433\) | \(-74814669657\) | \([2]\) | \(12288\) | \(0.76716\) |
Rank
sage: E.rank()
The elliptic curves in class 13923j have rank \(2\).
Complex multiplication
The elliptic curves in class 13923j do not have complex multiplication.Modular form 13923.2.a.j
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.