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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 50400.cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 50400.cm1 | 50400bq4 | \([0, 0, 0, -72903675, -239591960750]\) | \(7347751505995469192/72930375\) | \(425329947000000000\) | \([2]\) | \(2949120\) | \(2.9589\) | |
| 50400.cm2 | 50400bq3 | \([0, 0, 0, -6528675, -195898250]\) | \(5276930158229192/3050936350875\) | \(17793060798303000000000\) | \([2]\) | \(2949120\) | \(2.9589\) | |
| 50400.cm3 | 50400bq1 | \([0, 0, 0, -4559925, -3737679500]\) | \(14383655824793536/45209390625\) | \(32957645765625000000\) | \([2, 2]\) | \(1474560\) | \(2.6123\) | \(\Gamma_0(N)\)-optimal |
| 50400.cm4 | 50400bq2 | \([0, 0, 0, -2646300, -6898988000]\) | \(-43927191786304/415283203125\) | \(-19375453125000000000000\) | \([2]\) | \(2949120\) | \(2.9589\) |
Rank
sage: E.rank()
The elliptic curves in class 50400.cm have rank \(1\).
Complex multiplication
The elliptic curves in class 50400.cm do not have complex multiplication.Modular form 50400.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
