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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 371943ba
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 371943.ba5 | 371943ba1 | \([1, -1, 0, -62478, -8606849]\) | \(-1532808577/938223\) | \(-16509241929517623\) | \([2]\) | \(2359296\) | \(1.8139\) | \(\Gamma_0(N)\)-optimal |
| 371943.ba4 | 371943ba2 | \([1, -1, 0, -1115883, -453354440]\) | \(8732907467857/1656369\) | \(29145945628654569\) | \([2, 2]\) | \(4718592\) | \(2.1604\) | |
| 371943.ba3 | 371943ba3 | \([1, -1, 0, -1232928, -352344605]\) | \(11779205551777/3763454409\) | \(66222826906706364609\) | \([2, 2]\) | \(9437184\) | \(2.5070\) | |
| 371943.ba1 | 371943ba4 | \([1, -1, 0, -17853318, -29030850959]\) | \(35765103905346817/1287\) | \(22646422399887\) | \([2]\) | \(9437184\) | \(2.5070\) | |
| 371943.ba2 | 371943ba5 | \([1, -1, 0, -7826463, 8162546494]\) | \(3013001140430737/108679952667\) | \(1912363725327589515267\) | \([2]\) | \(18874368\) | \(2.8536\) | |
| 371943.ba6 | 371943ba6 | \([1, -1, 0, 3487887, -2404010804]\) | \(266679605718863/296110251723\) | \(-5210441210144464131123\) | \([2]\) | \(18874368\) | \(2.8536\) |
Rank
sage: E.rank()
The elliptic curves in class 371943ba have rank \(1\).
Complex multiplication
The elliptic curves in class 371943ba do not have complex multiplication.Modular form 371943.2.a.ba
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.
