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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 414.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414.d1 | 414b2 | \([1, -1, 1, -284, -1767]\) | \(3463512697/3174\) | \(2313846\) | \([2]\) | \(128\) | \(0.14475\) | |
414.d2 | 414b1 | \([1, -1, 1, -14, -39]\) | \(-389017/828\) | \(-603612\) | \([2]\) | \(64\) | \(-0.20182\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 414.d have rank \(0\).
Complex multiplication
The elliptic curves in class 414.d 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 LMFDB 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 LMFDB labels.