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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1005.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1005.a1 | 1005b2 | \([0, 1, 1, -3001, -70904]\) | \(-2989967081734144/380653171875\) | \(-380653171875\) | \([]\) | \(1440\) | \(0.95590\) | |
1005.a2 | 1005b1 | \([0, 1, 1, 239, 295]\) | \(1503484706816/890163675\) | \(-890163675\) | \([3]\) | \(480\) | \(0.40659\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1005.a have rank \(1\).
Complex multiplication
The elliptic curves in class 1005.a do not have complex multiplication.Modular form 1005.2.a.a
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.