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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 25410a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25410.e2 | 25410a1 | \([1, 1, 0, -2700238, -121338960332]\) | \(-923412886970939/2696845197312000\) | \(-6359019905986269806592000\) | \([2]\) | \(6589440\) | \(3.4383\) | \(\Gamma_0(N)\)-optimal |
25410.e1 | 25410a2 | \([1, 1, 0, -384324558, -2863614998988]\) | \(2662465301927918953019/38569862016000000\) | \(90945717082815805056000000\) | \([2]\) | \(13178880\) | \(3.7848\) |
Rank
sage: E.rank()
The elliptic curves in class 25410a have rank \(1\).
Complex multiplication
The elliptic curves in class 25410a do not have complex multiplication.Modular form 25410.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.