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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 224400cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 224400.ht3 | 224400cm1 | \([0, 1, 0, -17559008, -47937168012]\) | \(-9354997870579612441/10093752054144000\) | \(-646000131465216000000000\) | \([2]\) | \(26542080\) | \(3.2636\) | \(\Gamma_0(N)\)-optimal |
| 224400.ht2 | 224400cm2 | \([0, 1, 0, -331991008, -2327569168012]\) | \(63229930193881628103961/26218934428500000\) | \(1678011803424000000000000\) | \([2]\) | \(53084160\) | \(3.6102\) | |
| 224400.ht4 | 224400cm3 | \([0, 1, 0, 147170992, 865724771988]\) | \(5508208700580085578359/8246033269590589440\) | \(-527746129253797724160000000\) | \([2]\) | \(79626240\) | \(3.8130\) | |
| 224400.ht1 | 224400cm4 | \([0, 1, 0, -966941008, 8697932131988]\) | \(1562225332123379392365961/393363080510106009600\) | \(25175237152646784614400000000\) | \([2]\) | \(159252480\) | \(4.1595\) |
Rank
sage: E.rank()
The elliptic curves in class 224400cm have rank \(0\).
Complex multiplication
The elliptic curves in class 224400cm do not have complex multiplication.Modular form 224400.2.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 Cremona 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 Cremona labels.
