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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 363090bn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 363090.bn7 | 363090bn1 | \([1, 1, 0, -15806837, -23763252771]\) | \(3712533999213317890249/76090919904090000\) | \(8952020635796284410000\) | \([2]\) | \(31850496\) | \(3.0028\) | \(\Gamma_0(N)\)-optimal |
| 363090.bn6 | 363090bn2 | \([1, 1, 0, -33667337, 39780834129]\) | \(35872512095393194378249/14944558319037792900\) | \(1758212341676477296892100\) | \([2, 2]\) | \(63700992\) | \(3.3493\) | |
| 363090.bn5 | 363090bn3 | \([1, 1, 0, -147637172, 680359903056]\) | \(3024980849878413455018809/50557689000000000000\) | \(5948061553161000000000000\) | \([2]\) | \(95551488\) | \(3.5521\) | |
| 363090.bn4 | 363090bn4 | \([1, 1, 0, -464766887, 3854925631719]\) | \(94371532824107026279203049/40995077600666342790\) | \(4823029884640794562900710\) | \([2]\) | \(127401984\) | \(3.6959\) | |
| 363090.bn8 | 363090bn5 | \([1, 1, 0, 111664213, 291814808139]\) | \(1308812680909424992398551/1070002284841633041990\) | \(-125884698809333285757081510\) | \([2]\) | \(127401984\) | \(3.6959\) | |
| 363090.bn2 | 363090bn6 | \([1, 1, 0, -2352637172, 43920850903056]\) | \(12240533203187013248735018809/3506282465049000000\) | \(412510625730549801000000\) | \([2, 2]\) | \(191102976\) | \(3.8986\) | |
| 363090.bn1 | 363090bn7 | \([1, 1, 0, -37642192172, 2810981916364056]\) | \(50137213659805457275731367898809/4113897879000\) | \(483995971566471000\) | \([2]\) | \(382205952\) | \(4.2452\) | |
| 363090.bn3 | 363090bn8 | \([1, 1, 0, -2343082172, 44295309442056]\) | \(-12091997009671629064982138809/207252595706436249879000\) | \(-24383060632266518362014471000\) | \([2]\) | \(382205952\) | \(4.2452\) |
Rank
sage: E.rank()
The elliptic curves in class 363090bn have rank \(1\).
Complex multiplication
The elliptic curves in class 363090bn do not have complex multiplication.Modular form 363090.2.a.bn
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.
