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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 62400.cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 62400.cm1 | 62400z6 | \([0, -1, 0, -14424033, -21080412063]\) | \(81025909800741361/11088090\) | \(45416816640000000\) | \([2]\) | \(2359296\) | \(2.6085\) | |
| 62400.cm2 | 62400z4 | \([0, -1, 0, -1352033, 605075937]\) | \(66730743078481/60937500\) | \(249600000000000000\) | \([2]\) | \(1179648\) | \(2.2619\) | |
| 62400.cm3 | 62400z3 | \([0, -1, 0, -904033, -327212063]\) | \(19948814692561/231344100\) | \(947585433600000000\) | \([2, 2]\) | \(1179648\) | \(2.2619\) | |
| 62400.cm4 | 62400z5 | \([0, -1, 0, -184033, -834812063]\) | \(-168288035761/73415764890\) | \(-300710972989440000000\) | \([4]\) | \(2359296\) | \(2.6085\) | |
| 62400.cm5 | 62400z2 | \([0, -1, 0, -104033, 4787937]\) | \(30400540561/15210000\) | \(62300160000000000\) | \([2, 2]\) | \(589824\) | \(1.9154\) | |
| 62400.cm6 | 62400z1 | \([0, -1, 0, 23967, 563937]\) | \(371694959/249600\) | \(-1022361600000000\) | \([2]\) | \(294912\) | \(1.5688\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 62400.cm have rank \(2\).
Complex multiplication
The elliptic curves in class 62400.cm do not have complex multiplication.Modular form 62400.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
