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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 493680b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
493680.b1 | 493680b1 | \([0, -1, 0, -6376, -189440]\) | \(21035785676/541875\) | \(738545280000\) | \([2]\) | \(1179648\) | \(1.0594\) | \(\Gamma_0(N)\)-optimal |
493680.b2 | 493680b2 | \([0, -1, 0, 1104, -614304]\) | \(54541802/59765625\) | \(-162914400000000\) | \([2]\) | \(2359296\) | \(1.4060\) |
Rank
sage: E.rank()
The elliptic curves in class 493680b have rank \(2\).
Complex multiplication
The elliptic curves in class 493680b do not have complex multiplication.Modular form 493680.2.a.b
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.