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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 5025.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5025.b1 | 5025e1 | \([1, 0, 0, -7438, -228133]\) | \(2912566550041/254390625\) | \(3974853515625\) | \([2]\) | \(8640\) | \(1.1581\) | \(\Gamma_0(N)\)-optimal |
5025.b2 | 5025e2 | \([1, 0, 0, 8187, -1056258]\) | \(3883959939959/33133870125\) | \(-517716720703125\) | \([2]\) | \(17280\) | \(1.5047\) |
Rank
sage: E.rank()
The elliptic curves in class 5025.b have rank \(1\).
Complex multiplication
The elliptic curves in class 5025.b do not have complex multiplication.Modular form 5025.2.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.