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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 5712.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5712.b1 | 5712n1 | \([0, -1, 0, -2997, -63639]\) | \(-11632923639808/318495051\) | \(-81534733056\) | \([]\) | \(6912\) | \(0.87493\) | \(\Gamma_0(N)\)-optimal |
| 5712.b2 | 5712n2 | \([0, -1, 0, 13323, -265191]\) | \(1021544365555712/705905647251\) | \(-180711845696256\) | \([]\) | \(20736\) | \(1.4242\) |
Rank
sage: E.rank()
The elliptic curves in class 5712.b have rank \(1\).
Complex multiplication
The elliptic curves in class 5712.b do not have complex multiplication.Modular form 5712.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
