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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 323400.cm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
323400.cm1 | 323400cm4 | \([0, -1, 0, -784284608, 5994647023212]\) | \(14171198121996897746/4077720290568771\) | \(15351670862884010860128000000\) | \([2]\) | \(283115520\) | \(4.1148\) | |
323400.cm2 | 323400cm2 | \([0, -1, 0, -719065608, 7420986553212]\) | \(21843440425782779332/3100814593569\) | \(5836923777900788496000000\) | \([2, 2]\) | \(141557760\) | \(3.7682\) | |
323400.cm3 | 323400cm1 | \([0, -1, 0, -719041108, 7421517566212]\) | \(87364831012240243408/1760913\) | \(828678614148000000\) | \([2]\) | \(70778880\) | \(3.4217\) | \(\Gamma_0(N)\)-optimal |
323400.cm4 | 323400cm3 | \([0, -1, 0, -654238608, 8813340859212]\) | \(-8226100326647904626/4152140742401883\) | \(-15631846598490852258144000000\) | \([2]\) | \(283115520\) | \(4.1148\) |
Rank
sage: E.rank()
The elliptic curves in class 323400.cm have rank \(0\).
Complex multiplication
The elliptic curves in class 323400.cm do not have complex multiplication.Modular form 323400.2.a.cm
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.