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SageMath
E = EllipticCurve("il1")
E.isogeny_class()
Elliptic curves in class 177870il
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 177870.k7 | 177870il1 | \([1, 1, 0, 1244967, 403409637]\) | \(1023887723039/928972800\) | \(-193618722013942579200\) | \([2]\) | \(7864320\) | \(2.5800\) | \(\Gamma_0(N)\)-optimal |
| 177870.k6 | 177870il2 | \([1, 1, 0, -6344153, 3604500453]\) | \(135487869158881/51438240000\) | \(10720880408389203360000\) | \([2, 2]\) | \(15728640\) | \(2.9266\) | |
| 177870.k4 | 177870il3 | \([1, 1, 0, -89350153, 324953928853]\) | \(378499465220294881/120530818800\) | \(25121320128371986873200\) | \([2]\) | \(31457280\) | \(3.2732\) | |
| 177870.k5 | 177870il4 | \([1, 1, 0, -44764073, -112723333323]\) | \(47595748626367201/1215506250000\) | \(253338705638055056250000\) | \([2, 2]\) | \(31457280\) | \(3.2732\) | |
| 177870.k8 | 177870il5 | \([1, 1, 0, 7529707, -360271629087]\) | \(226523624554079/269165039062500\) | \(-56100018078154907226562500\) | \([2]\) | \(62914560\) | \(3.6198\) | |
| 177870.k2 | 177870il6 | \([1, 1, 0, -711776573, -7309388000823]\) | \(191342053882402567201/129708022500\) | \(27034054766087564002500\) | \([2, 2]\) | \(62914560\) | \(3.6198\) | |
| 177870.k3 | 177870il7 | \([1, 1, 0, -707329823, -7405218131373]\) | \(-187778242790732059201/4984939585440150\) | \(-1038972972997309033661173350\) | \([2]\) | \(125829120\) | \(3.9663\) | |
| 177870.k1 | 177870il8 | \([1, 1, 0, -11388423323, -467786756340273]\) | \(783736670177727068275201/360150\) | \(75063320189053350\) | \([2]\) | \(125829120\) | \(3.9663\) |
Rank
sage: E.rank()
The elliptic curves in class 177870il have rank \(1\).
Complex multiplication
The elliptic curves in class 177870il do not have complex multiplication.Modular form 177870.2.a.il
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
