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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 414d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414.b2 | 414d1 | \([1, -1, 1, -92, 415]\) | \(-116930169/23552\) | \(-17169408\) | \([2]\) | \(160\) | \(0.11082\) | \(\Gamma_0(N)\)-optimal |
414.b1 | 414d2 | \([1, -1, 1, -1532, 23455]\) | \(545138290809/16928\) | \(12340512\) | \([2]\) | \(320\) | \(0.45740\) |
Rank
sage: E.rank()
The elliptic curves in class 414d have rank \(1\).
Complex multiplication
The elliptic curves in class 414d do not have complex multiplication.Modular form 414.2.a.d
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.