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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 858.h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
858.h1 | 858e3 | \([1, 1, 1, -13267, -589879]\) | \(258252149810350513/1938176193096\) | \(1938176193096\) | \([2]\) | \(2304\) | \(1.1879\) | |
858.h2 | 858e2 | \([1, 1, 1, -1387, 4121]\) | \(295102348042033/161237583936\) | \(161237583936\) | \([2, 2]\) | \(1152\) | \(0.84128\) | |
858.h3 | 858e1 | \([1, 1, 1, -1067, 12953]\) | \(134351465835313/205590528\) | \(205590528\) | \([4]\) | \(576\) | \(0.49471\) | \(\Gamma_0(N)\)-optimal |
858.h4 | 858e4 | \([1, 1, 1, 5373, 39273]\) | \(17154149157653327/10519679024712\) | \(-10519679024712\) | \([2]\) | \(2304\) | \(1.1879\) |
Rank
sage: E.rank()
The elliptic curves in class 858.h have rank \(0\).
Complex multiplication
The elliptic curves in class 858.h do not have complex multiplication.Modular form 858.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.