Show commands:
SageMath
E = EllipticCurve("qt1")
E.isogeny_class()
Elliptic curves in class 235200qt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 235200.qt6 | 235200qt1 | \([0, 1, 0, 782367, 373012863]\) | \(109902239/188160\) | \(-90672479600640000000\) | \([2]\) | \(7077888\) | \(2.5144\) | \(\Gamma_0(N)\)-optimal |
| 235200.qt5 | 235200qt2 | \([0, 1, 0, -5489633, 3853972863]\) | \(37966934881/8643600\) | \(4165267031654400000000\) | \([2, 2]\) | \(14155776\) | \(2.8610\) | |
| 235200.qt2 | 235200qt3 | \([0, 1, 0, -82321633, 287440884863]\) | \(128031684631201/9922500\) | \(4781556541440000000000\) | \([2, 2]\) | \(28311552\) | \(3.2076\) | |
| 235200.qt4 | 235200qt4 | \([0, 1, 0, -29009633, -56851147137]\) | \(5602762882081/345888060\) | \(166680102383370240000000\) | \([2]\) | \(28311552\) | \(3.2076\) | |
| 235200.qt1 | 235200qt5 | \([0, 1, 0, -1317121633, 18398252484863]\) | \(524388516989299201/3150\) | \(1517954457600000000\) | \([2]\) | \(56623104\) | \(3.5542\) | |
| 235200.qt3 | 235200qt6 | \([0, 1, 0, -76833633, 327420964863]\) | \(-104094944089921/35880468750\) | \(-17290449993600000000000000\) | \([2]\) | \(56623104\) | \(3.5542\) |
Rank
sage: E.rank()
The elliptic curves in class 235200qt have rank \(1\).
Complex multiplication
The elliptic curves in class 235200qt do not have complex multiplication.Modular form 235200.2.a.qt
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
