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SageMath
E = EllipticCurve("gt1")
E.isogeny_class()
Elliptic curves in class 100800.gt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 100800.gt1 | 100800lv8 | \([0, 0, 0, -27659520300, -1770578063998000]\) | \(783736670177727068275201/360150\) | \(1075402137600000000\) | \([2]\) | \(75497472\) | \(4.1882\) | |
| 100800.gt2 | 100800lv6 | \([0, 0, 0, -1728720300, -27665272798000]\) | \(191342053882402567201/129708022500\) | \(387306079856640000000000\) | \([2, 2]\) | \(37748736\) | \(3.8416\) | |
| 100800.gt3 | 100800lv7 | \([0, 0, 0, -1717920300, -28028001598000]\) | \(-187778242790732059201/4984939585440150\) | \(-14884949843090920857600000000\) | \([2]\) | \(75497472\) | \(4.1882\) | |
| 100800.gt4 | 100800lv4 | \([0, 0, 0, -217008300, 1230107618000]\) | \(378499465220294881/120530818800\) | \(359903096443699200000000\) | \([2]\) | \(18874368\) | \(3.4950\) | |
| 100800.gt5 | 100800lv3 | \([0, 0, 0, -108720300, -426592798000]\) | \(47595748626367201/1215506250000\) | \(3629482214400000000000000\) | \([2, 2]\) | \(18874368\) | \(3.4950\) | |
| 100800.gt6 | 100800lv2 | \([0, 0, 0, -15408300, 13653218000]\) | \(135487869158881/51438240000\) | \(153593761628160000000000\) | \([2, 2]\) | \(9437184\) | \(3.1485\) | |
| 100800.gt7 | 100800lv1 | \([0, 0, 0, 3023700, 1524962000]\) | \(1023887723039/928972800\) | \(-2773897917235200000000\) | \([2]\) | \(4718592\) | \(2.8019\) | \(\Gamma_0(N)\)-optimal |
| 100800.gt8 | 100800lv5 | \([0, 0, 0, 18287700, -1363657822000]\) | \(226523624554079/269165039062500\) | \(-803722500000000000000000000\) | \([2]\) | \(37748736\) | \(3.8416\) |
Rank
sage: E.rank()
The elliptic curves in class 100800.gt have rank \(1\).
Complex multiplication
The elliptic curves in class 100800.gt do not have complex multiplication.Modular form 100800.2.a.gt
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
