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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 23184bi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 23184.bk5 | 23184bi1 | \([0, 0, 0, 18141, 1926178]\) | \(221115865823/664731648\) | \(-1984878065221632\) | \([2]\) | \(98304\) | \(1.6187\) | \(\Gamma_0(N)\)-optimal |
| 23184.bk4 | 23184bi2 | \([0, 0, 0, -166179, 22311970]\) | \(169967019783457/26337394944\) | \(78643039904464896\) | \([2, 2]\) | \(196608\) | \(1.9653\) | |
| 23184.bk3 | 23184bi3 | \([0, 0, 0, -730659, -218720990]\) | \(14447092394873377/1439452851984\) | \(4298183184778592256\) | \([2, 2]\) | \(393216\) | \(2.3119\) | |
| 23184.bk2 | 23184bi4 | \([0, 0, 0, -2550819, 1568035618]\) | \(614716917569296417/19093020912\) | \(57011454954897408\) | \([2]\) | \(393216\) | \(2.3119\) | |
| 23184.bk6 | 23184bi5 | \([0, 0, 0, 902301, -1057735838]\) | \(27207619911317663/177609314617308\) | \(-530338571698247811072\) | \([2]\) | \(786432\) | \(2.6584\) | |
| 23184.bk1 | 23184bi6 | \([0, 0, 0, -11395299, -14805815582]\) | \(54804145548726848737/637608031452\) | \(1903887380187168768\) | \([2]\) | \(786432\) | \(2.6584\) |
Rank
sage: E.rank()
The elliptic curves in class 23184bi have rank \(1\).
Complex multiplication
The elliptic curves in class 23184bi do not have complex multiplication.Modular form 23184.2.a.bi
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
