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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 242760cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 242760.cr4 | 242760cr1 | \([0, 1, 0, -50671, 4373330]\) | \(37256083456/525\) | \(202755579600\) | \([2]\) | \(655360\) | \(1.3098\) | \(\Gamma_0(N)\)-optimal |
| 242760.cr3 | 242760cr2 | \([0, 1, 0, -52116, 4109184]\) | \(2533446736/275625\) | \(1703146868640000\) | \([2, 2]\) | \(1310720\) | \(1.6564\) | |
| 242760.cr5 | 242760cr3 | \([0, 1, 0, 69264, 20519760]\) | \(1486779836/8203125\) | \(-202755579600000000\) | \([2]\) | \(2621440\) | \(2.0029\) | |
| 242760.cr2 | 242760cr4 | \([0, 1, 0, -196616, -29183616]\) | \(34008619684/4862025\) | \(120174043051238400\) | \([2, 2]\) | \(2621440\) | \(2.0029\) | |
| 242760.cr6 | 242760cr5 | \([0, 1, 0, 323584, -156944736]\) | \(75798394558/259416045\) | \(-12823905660712151040\) | \([2]\) | \(5242880\) | \(2.3495\) | |
| 242760.cr1 | 242760cr6 | \([0, 1, 0, -3028816, -2029849696]\) | \(62161150998242/1607445\) | \(79462020303267840\) | \([2]\) | \(5242880\) | \(2.3495\) |
Rank
sage: E.rank()
The elliptic curves in class 242760cr have rank \(0\).
Complex multiplication
The elliptic curves in class 242760cr do not have complex multiplication.Modular form 242760.2.a.cr
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
