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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 1290.h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1290.h1 | 1290g4 | \([1, 0, 1, -5183, -124342]\) | \(15393836938735081/2275690697640\) | \(2275690697640\) | \([2]\) | \(2880\) | \(1.0958\) | |
1290.h2 | 1290g3 | \([1, 0, 1, -4983, -135782]\) | \(13679527032530281/381633600\) | \(381633600\) | \([2]\) | \(1440\) | \(0.74920\) | |
1290.h3 | 1290g2 | \([1, 0, 1, -1358, 19118]\) | \(276670733768281/336980250\) | \(336980250\) | \([6]\) | \(960\) | \(0.54647\) | |
1290.h4 | 1290g1 | \([1, 0, 1, -108, 118]\) | \(137467988281/72562500\) | \(72562500\) | \([6]\) | \(480\) | \(0.19989\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1290.h have rank \(0\).
Complex multiplication
The elliptic curves in class 1290.h do not have complex multiplication.Modular form 1290.2.a.h
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.