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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 5054a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5054.a2 | 5054a1 | \([1, -1, 0, 56, -4704]\) | \(53261199/26353376\) | \(-9513568736\) | \([]\) | \(2520\) | \(0.59374\) | \(\Gamma_0(N)\)-optimal |
5054.a1 | 5054a2 | \([1, -1, 0, -75754, 8056692]\) | \(-133179212896925841/240518168576\) | \(-86827058855936\) | \([]\) | \(17640\) | \(1.5667\) |
Rank
sage: E.rank()
The elliptic curves in class 5054a have rank \(1\).
Complex multiplication
The elliptic curves in class 5054a do not have complex multiplication.Modular form 5054.2.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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.