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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 127449.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
127449.e1 | 127449bq2 | \([0, 0, 1, -2372979, -1408113702]\) | \(-1713910976512/1594323\) | \(-1374654151431932427\) | \([]\) | \(3234816\) | \(2.4033\) | |
127449.e2 | 127449bq1 | \([0, 0, 1, -6069, 197748]\) | \(-28672/3\) | \(-2586654306747\) | \([]\) | \(248832\) | \(1.1208\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 127449.e have rank \(1\).
Complex multiplication
The elliptic curves in class 127449.e do not have complex multiplication.Modular form 127449.2.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.
\(\left(\begin{array}{rr} 1 & 13 \\ 13 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.