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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 25857q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25857.k2 | 25857q1 | \([0, 0, 1, 1014, 59868]\) | \(32768/459\) | \(-1615103386299\) | \([]\) | \(37440\) | \(1.0233\) | \(\Gamma_0(N)\)-optimal |
25857.k1 | 25857q2 | \([0, 0, 1, -90246, 10440693]\) | \(-23100424192/14739\) | \(-51862764293379\) | \([]\) | \(112320\) | \(1.5726\) |
Rank
sage: E.rank()
The elliptic curves in class 25857q have rank \(1\).
Complex multiplication
The elliptic curves in class 25857q do not have complex multiplication.Modular form 25857.2.a.q
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.