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SageMath
E = EllipticCurve("gp1")
E.isogeny_class()
Elliptic curves in class 254800gp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
254800.gp2 | 254800gp1 | \([0, -1, 0, -143733, 21059837]\) | \(-43614208/91\) | \(-685187776000000\) | \([]\) | \(1492992\) | \(1.7323\) | \(\Gamma_0(N)\)-optimal |
254800.gp3 | 254800gp2 | \([0, -1, 0, 248267, 104163837]\) | \(224755712/753571\) | \(-5674039973056000000\) | \([]\) | \(4478976\) | \(2.2816\) | |
254800.gp1 | 254800gp3 | \([0, -1, 0, -2299733, -3307608163]\) | \(-178643795968/524596891\) | \(-3949971176272576000000\) | \([]\) | \(13436928\) | \(2.8309\) |
Rank
sage: E.rank()
The elliptic curves in class 254800gp have rank \(0\).
Complex multiplication
The elliptic curves in class 254800gp do not have complex multiplication.Modular form 254800.2.a.gp
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.