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SageMath
E = EllipticCurve("ch1")
E.isogeny_class()
Elliptic curves in class 11760ch
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 11760.bx7 | 11760ch1 | \([0, 1, 0, 164624, 19427540]\) | \(1023887723039/928972800\) | \(-447662984999731200\) | \([2]\) | \(147456\) | \(2.0742\) | \(\Gamma_0(N)\)-optimal |
| 11760.bx6 | 11760ch2 | \([0, 1, 0, -838896, 173166804]\) | \(135487869158881/51438240000\) | \(24787589110824960000\) | \([2, 2]\) | \(294912\) | \(2.4208\) | |
| 11760.bx5 | 11760ch3 | \([0, 1, 0, -5919216, -5421281580]\) | \(47595748626367201/1215506250000\) | \(585740676326400000000\) | \([2, 2]\) | \(589824\) | \(2.7674\) | |
| 11760.bx4 | 11760ch4 | \([0, 1, 0, -11814896, 15622984404]\) | \(378499465220294881/120530818800\) | \(58082632912900915200\) | \([4]\) | \(589824\) | \(2.7674\) | |
| 11760.bx2 | 11760ch5 | \([0, 1, 0, -94119216, -351482801580]\) | \(191342053882402567201/129708022500\) | \(62505038393763840000\) | \([2, 2]\) | \(1179648\) | \(3.1140\) | |
| 11760.bx8 | 11760ch6 | \([0, 1, 0, 995664, -17323173036]\) | \(226523624554079/269165039062500\) | \(-129708022500000000000000\) | \([2]\) | \(1179648\) | \(3.1140\) | |
| 11760.bx1 | 11760ch7 | \([0, 1, 0, -1505907216, -22493401078380]\) | \(783736670177727068275201/360150\) | \(173552792985600\) | \([2]\) | \(2359296\) | \(3.4605\) | |
| 11760.bx3 | 11760ch8 | \([0, 1, 0, -93531216, -356090604780]\) | \(-187778242790732059201/4984939585440150\) | \(-2402194052249387857305600\) | \([2]\) | \(2359296\) | \(3.4605\) |
Rank
sage: E.rank()
The elliptic curves in class 11760ch have rank \(0\).
Complex multiplication
The elliptic curves in class 11760ch do not have complex multiplication.Modular form 11760.2.a.ch
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
