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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 25857r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25857.n2 | 25857r1 | \([1, -1, 0, 14418, 680967]\) | \(42875/51\) | \(-394264682188767\) | \([2]\) | \(79872\) | \(1.4869\) | \(\Gamma_0(N)\)-optimal |
25857.n1 | 25857r2 | \([1, -1, 0, -84447, 6553548]\) | \(8615125/2601\) | \(20107498791627117\) | \([2]\) | \(159744\) | \(1.8335\) |
Rank
sage: E.rank()
The elliptic curves in class 25857r have rank \(1\).
Complex multiplication
The elliptic curves in class 25857r do not have complex multiplication.Modular form 25857.2.a.r
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.