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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 43758.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
43758.b1 | 43758c1 | \([1, -1, 0, -137553, 10733021]\) | \(14623266529962819/5950368579584\) | \(117121104751951872\) | \([2]\) | \(466944\) | \(1.9729\) | \(\Gamma_0(N)\)-optimal |
43758.b2 | 43758c2 | \([1, -1, 0, 449967, 77827805]\) | \(511886728354194621/429557271832832\) | \(-8454975781485632256\) | \([2]\) | \(933888\) | \(2.3195\) |
Rank
sage: E.rank()
The elliptic curves in class 43758.b have rank \(2\).
Complex multiplication
The elliptic curves in class 43758.b do not have complex multiplication.Modular form 43758.2.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 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.