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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 40128a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
40128.q2 | 40128a1 | \([0, -1, 0, -1453, -22505]\) | \(-5304438784000/497763387\) | \(-31856856768\) | \([]\) | \(25920\) | \(0.75744\) | \(\Gamma_0(N)\)-optimal |
40128.q1 | 40128a2 | \([0, -1, 0, -120253, -16010609]\) | \(-3004935183806464000/2037123\) | \(-130375872\) | \([]\) | \(77760\) | \(1.3067\) |
Rank
sage: E.rank()
The elliptic curves in class 40128a have rank \(1\).
Complex multiplication
The elliptic curves in class 40128a do not have complex multiplication.Modular form 40128.2.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 Cremona 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 Cremona labels.