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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 714.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
714.e1 | 714f4 | \([1, 1, 1, -5079, 137205]\) | \(14489843500598257/6246072\) | \(6246072\) | \([2]\) | \(768\) | \(0.64655\) | |
714.e2 | 714f3 | \([1, 1, 1, -679, -3883]\) | \(34623662831857/14438442312\) | \(14438442312\) | \([2]\) | \(768\) | \(0.64655\) | |
714.e3 | 714f2 | \([1, 1, 1, -319, 2021]\) | \(3590714269297/73410624\) | \(73410624\) | \([2, 2]\) | \(384\) | \(0.29998\) | |
714.e4 | 714f1 | \([1, 1, 1, 1, 101]\) | \(103823/4386816\) | \(-4386816\) | \([4]\) | \(192\) | \(-0.046597\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 714.e have rank \(1\).
Complex multiplication
The elliptic curves in class 714.e do not have complex multiplication.Modular form 714.2.a.e
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.