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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 4624.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4624.e1 | 4624b2 | \([0, -1, 0, -106448, 13392656]\) | \(1098500\) | \(121433985532928\) | \([2]\) | \(17408\) | \(1.6264\) | |
4624.e2 | 4624b1 | \([0, -1, 0, -8188, 107904]\) | \(2000\) | \(30358496383232\) | \([2]\) | \(8704\) | \(1.2798\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4624.e have rank \(1\).
Complex multiplication
The elliptic curves in class 4624.e do not have complex multiplication.Modular form 4624.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.