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SageMath
E = EllipticCurve("gc1")
E.isogeny_class()
Elliptic curves in class 106470gc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 106470.fp7 | 106470gc1 | \([1, -1, 1, 319378, -52428931]\) | \(1023887723039/928972800\) | \(-3268817244138700800\) | \([2]\) | \(2359296\) | \(2.2399\) | \(\Gamma_0(N)\)-optimal |
| 106470.fp6 | 106470gc2 | \([1, -1, 1, -1627502, -468282499]\) | \(135487869158881/51438240000\) | \(180997986076820640000\) | \([2, 2]\) | \(4718592\) | \(2.5865\) | |
| 106470.fp5 | 106470gc3 | \([1, -1, 1, -11483582, 14647001789]\) | \(47595748626367201/1215506250000\) | \(4277055033644006250000\) | \([2, 2]\) | \(9437184\) | \(2.9331\) | |
| 106470.fp4 | 106470gc4 | \([1, -1, 1, -22921502, -42221557699]\) | \(378499465220294881/120530818800\) | \(424117066660721506800\) | \([2]\) | \(9437184\) | \(2.9331\) | |
| 106470.fp8 | 106470gc5 | \([1, -1, 1, 1931638, 46811333261]\) | \(226523624554079/269165039062500\) | \(-947122801880493164062500\) | \([2]\) | \(18874368\) | \(3.2796\) | |
| 106470.fp2 | 106470gc6 | \([1, -1, 1, -182596082, 949742591789]\) | \(191342053882402567201/129708022500\) | \(456409294923522622500\) | \([2, 2]\) | \(18874368\) | \(3.2796\) | |
| 106470.fp3 | 106470gc7 | \([1, -1, 1, -181455332, 962194106189]\) | \(-187778242790732059201/4984939585440150\) | \(-17540725065229454251404150\) | \([2]\) | \(37748736\) | \(3.6262\) | |
| 106470.fp1 | 106470gc8 | \([1, -1, 1, -2921536832, 60781355487389]\) | \(783736670177727068275201/360150\) | \(1267275565524150\) | \([2]\) | \(37748736\) | \(3.6262\) |
Rank
sage: E.rank()
The elliptic curves in class 106470gc have rank \(1\).
Complex multiplication
The elliptic curves in class 106470gc do not have complex multiplication.Modular form 106470.2.a.gc
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
