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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 414050.ek
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414050.ek1 | 414050ek2 | \([1, 0, 0, -37889888, -89701764608]\) | \(553463785/512\) | \(5565118670001800000000\) | \([]\) | \(52254720\) | \(3.0957\) | |
414050.ek2 | 414050ek1 | \([1, 0, 0, -1660513, 690526017]\) | \(46585/8\) | \(86954979218778125000\) | \([]\) | \(17418240\) | \(2.5464\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 414050.ek have rank \(1\).
Complex multiplication
The elliptic curves in class 414050.ek do not have complex multiplication.Modular form 414050.2.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 LMFDB 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 LMFDB labels.