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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 20400.x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20400.x1 | 20400bt7 | \([0, -1, 0, -45388008, 117710038512]\) | \(161572377633716256481/914742821250\) | \(58543540560000000000\) | \([2]\) | \(1179648\) | \(2.9846\) | |
| 20400.x2 | 20400bt3 | \([0, -1, 0, -8704008, -9880969488]\) | \(1139466686381936641/4080\) | \(261120000000\) | \([2]\) | \(294912\) | \(2.2915\) | |
| 20400.x3 | 20400bt5 | \([0, -1, 0, -2888008, 1770038512]\) | \(41623544884956481/2962701562500\) | \(189612900000000000000\) | \([2, 2]\) | \(589824\) | \(2.6380\) | |
| 20400.x4 | 20400bt4 | \([0, -1, 0, -576008, -135049488]\) | \(330240275458561/67652010000\) | \(4329728640000000000\) | \([2, 2]\) | \(294912\) | \(2.2915\) | |
| 20400.x5 | 20400bt2 | \([0, -1, 0, -544008, -154249488]\) | \(278202094583041/16646400\) | \(1065369600000000\) | \([2, 2]\) | \(147456\) | \(1.9449\) | |
| 20400.x6 | 20400bt1 | \([0, -1, 0, -32008, -2697488]\) | \(-56667352321/16711680\) | \(-1069547520000000\) | \([2]\) | \(73728\) | \(1.5983\) | \(\Gamma_0(N)\)-optimal |
| 20400.x7 | 20400bt6 | \([0, -1, 0, 1223992, -811849488]\) | \(3168685387909439/6278181696900\) | \(-401803628601600000000\) | \([2]\) | \(589824\) | \(2.6380\) | |
| 20400.x8 | 20400bt8 | \([0, -1, 0, 2619992, 7718678512]\) | \(31077313442863199/420227050781250\) | \(-26894531250000000000000\) | \([2]\) | \(1179648\) | \(2.9846\) |
Rank
sage: E.rank()
The elliptic curves in class 20400.x have rank \(0\).
Complex multiplication
The elliptic curves in class 20400.x do not have complex multiplication.Modular form 20400.2.a.x
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 16 & 2 & 4 & 8 & 16 & 8 & 4 \\ 16 & 1 & 8 & 4 & 2 & 4 & 8 & 16 \\ 2 & 8 & 1 & 2 & 4 & 8 & 4 & 2 \\ 4 & 4 & 2 & 1 & 2 & 4 & 2 & 4 \\ 8 & 2 & 4 & 2 & 1 & 2 & 4 & 8 \\ 16 & 4 & 8 & 4 & 2 & 1 & 8 & 16 \\ 8 & 8 & 4 & 2 & 4 & 8 & 1 & 8 \\ 4 & 16 & 2 & 4 & 8 & 16 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
