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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 258570ek
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
258570.ek1 | 258570ek1 | \([1, -1, 1, -999981482, -12171024079639]\) | \(31427652507069423952801/654426190080\) | \(2302758073379000090880\) | \([2]\) | \(65372160\) | \(3.6284\) | \(\Gamma_0(N)\)-optimal |
258570.ek2 | 258570ek2 | \([1, -1, 1, -998886362, -12199012718551]\) | \(-31324512477868037557921/143427974919699600\) | \(-504686291901557443494195600\) | \([2]\) | \(130744320\) | \(3.9750\) |
Rank
sage: E.rank()
The elliptic curves in class 258570ek have rank \(1\).
Complex multiplication
The elliptic curves in class 258570ek do not have complex multiplication.Modular form 258570.2.a.ek
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.