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SageMath
E = EllipticCurve("gt1")
E.isogeny_class()
Elliptic curves in class 141120.gt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 141120.gt1 | 141120el7 | \([0, 0, 0, -54212659788, 4858466207610512]\) | \(783736670177727068275201/360150\) | \(8097279109536153600\) | \([2]\) | \(150994944\) | \(4.3564\) | |
| 141120.gt2 | 141120el5 | \([0, 0, 0, -3388291788, 75913508557712]\) | \(191342053882402567201/129708022500\) | \(2916235071299445719040000\) | \([2, 2]\) | \(75497472\) | \(4.0098\) | |
| 141120.gt3 | 141120el8 | \([0, 0, 0, -3367123788, 76908836384912]\) | \(-187778242790732059201/4984939585440150\) | \(-112076765701747439870450073600\) | \([2]\) | \(150994944\) | \(4.3564\) | |
| 141120.gt4 | 141120el4 | \([0, 0, 0, -425336268, -3375415303792]\) | \(378499465220294881/120530818800\) | \(2709903321184305099571200\) | \([2]\) | \(37748736\) | \(3.6633\) | |
| 141120.gt5 | 141120el3 | \([0, 0, 0, -213091788, 1170570637712]\) | \(47595748626367201/1215506250000\) | \(27328316994684518400000000\) | \([2, 2]\) | \(37748736\) | \(3.6633\) | |
| 141120.gt6 | 141120el2 | \([0, 0, 0, -30200268, -37464430192]\) | \(135487869158881/51438240000\) | \(1156489757554649333760000\) | \([2, 2]\) | \(18874368\) | \(3.3167\) | |
| 141120.gt7 | 141120el1 | \([0, 0, 0, 5926452, -4184495728]\) | \(1023887723039/928972800\) | \(-20886164228147458867200\) | \([2]\) | \(9437184\) | \(2.9701\) | \(\Gamma_0(N)\)-optimal |
| 141120.gt8 | 141120el6 | \([0, 0, 0, 35843892, 3741877063568]\) | \(226523624554079/269165039062500\) | \(-6051657497760000000000000000\) | \([2]\) | \(75497472\) | \(4.0098\) |
Rank
sage: E.rank()
The elliptic curves in class 141120.gt have rank \(0\).
Complex multiplication
The elliptic curves in class 141120.gt do not have complex multiplication.Modular form 141120.2.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.
