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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 40362.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
40362.d1 | 40362i2 | \([1, 1, 0, -568451, -165146211]\) | \(22889370414457/8718192\) | \(7737427491664752\) | \([2]\) | \(737280\) | \(2.0149\) | |
40362.d2 | 40362i1 | \([1, 1, 0, -30291, -3375315]\) | \(-3463512697/3499776\) | \(-3106064082675456\) | \([2]\) | \(368640\) | \(1.6683\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 40362.d have rank \(1\).
Complex multiplication
The elliptic curves in class 40362.d do not have complex multiplication.Modular form 40362.2.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.