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SageMath
E = EllipticCurve("es1")
E.isogeny_class()
Elliptic curves in class 254898es
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
254898.es2 | 254898es1 | \([1, -1, 1, 18586, -66726411]\) | \(189/512\) | \(-1923585849106822656\) | \([]\) | \(4644864\) | \(2.1873\) | \(\Gamma_0(N)\)-optimal |
254898.es1 | 254898es2 | \([1, -1, 1, -11876654, -15752582891]\) | \(-67645179/8\) | \(-21910845062482401816\) | \([]\) | \(13934592\) | \(2.7366\) |
Rank
sage: E.rank()
The elliptic curves in class 254898es have rank \(0\).
Complex multiplication
The elliptic curves in class 254898es do not have complex multiplication.Modular form 254898.2.a.es
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.