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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 32400.cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 32400.cm1 | 32400da3 | \([0, 0, 0, -430875, -108861750]\) | \(-189613868625/128\) | \(-5971968000000\) | \([]\) | \(145152\) | \(1.7672\) | |
| 32400.cm2 | 32400da4 | \([0, 0, 0, -340875, -155607750]\) | \(-1159088625/2097152\) | \(-7925422620672000000\) | \([]\) | \(435456\) | \(2.3165\) | |
| 32400.cm3 | 32400da2 | \([0, 0, 0, -16875, 884250]\) | \(-140625/8\) | \(-30233088000000\) | \([]\) | \(62208\) | \(1.3436\) | |
| 32400.cm4 | 32400da1 | \([0, 0, 0, 1125, 2250]\) | \(3375/2\) | \(-93312000000\) | \([]\) | \(20736\) | \(0.79428\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 32400.cm have rank \(1\).
Complex multiplication
The elliptic curves in class 32400.cm do not have complex multiplication.Modular form 32400.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 21 & 7 \\ 3 & 1 & 7 & 21 \\ 21 & 7 & 1 & 3 \\ 7 & 21 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
