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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 103933.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
103933.b1 | 103933b3 | \([0, -1, 1, -5262193, -4644450356]\) | \(727057727488000/37\) | \(820081361773\) | \([]\) | \(906984\) | \(2.2072\) | |
103933.b2 | 103933b2 | \([0, -1, 1, -65543, -6232365]\) | \(1404928000/50653\) | \(1122691384267237\) | \([]\) | \(302328\) | \(1.6579\) | |
103933.b3 | 103933b1 | \([0, -1, 1, -9363, 349122]\) | \(4096000/37\) | \(820081361773\) | \([]\) | \(100776\) | \(1.1086\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 103933.b have rank \(0\).
Complex multiplication
The elliptic curves in class 103933.b do not have complex multiplication.Modular form 103933.2.a.b
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.