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SageMath
E = EllipticCurve("ll1")
E.isogeny_class()
Elliptic curves in class 262080ll
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 262080.ll7 | 262080ll1 | \([0, 0, 0, -14817612, -21836400784]\) | \(1882742462388824401/11650189824000\) | \(2226385946331316224000\) | \([2]\) | \(14155776\) | \(2.9344\) | \(\Gamma_0(N)\)-optimal |
| 262080.ll6 | 262080ll2 | \([0, 0, 0, -23849292, 7928403824]\) | \(7850236389974007121/4400862921000000\) | \(841018001171152896000000\) | \([2, 2]\) | \(28311552\) | \(3.2809\) | |
| 262080.ll5 | 262080ll3 | \([0, 0, 0, -91540812, 322423500656]\) | \(443915739051786565201/21894701746029840\) | \(4184142662298698608803840\) | \([2]\) | \(42467328\) | \(3.4837\) | |
| 262080.ll4 | 262080ll4 | \([0, 0, 0, -285929292, 1857898707824]\) | \(13527956825588849127121/25701087819771000\) | \(4911554368795589738496000\) | \([2]\) | \(56623104\) | \(3.6275\) | |
| 262080.ll8 | 262080ll5 | \([0, 0, 0, 93723828, 62905594736]\) | \(476437916651992691759/284661685546875000\) | \(-54399695261184000000000000\) | \([2]\) | \(56623104\) | \(3.6275\) | |
| 262080.ll2 | 262080ll6 | \([0, 0, 0, -1446857292, 21182912634224]\) | \(1752803993935029634719121/4599740941532100\) | \(879024182755826309529600\) | \([2, 2]\) | \(84934656\) | \(3.8302\) | |
| 262080.ll1 | 262080ll7 | \([0, 0, 0, -23149702092, 1355708201662064]\) | \(7179471593960193209684686321/49441793310\) | \(9448473840317890560\) | \([2]\) | \(169869312\) | \(4.1768\) | |
| 262080.ll3 | 262080ll8 | \([0, 0, 0, -1429076172, 21728928154736]\) | \(-1688971789881664420008241/89901485966373558750\) | \(-17180441514996223004835840000\) | \([2]\) | \(169869312\) | \(4.1768\) |
Rank
sage: E.rank()
The elliptic curves in class 262080ll have rank \(0\).
Complex multiplication
The elliptic curves in class 262080ll do not have complex multiplication.Modular form 262080.2.a.ll
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
