Show commands:
SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 64680.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 64680.f1 | 64680bh6 | \([0, -1, 0, -82559136, 288751969836]\) | \(258286045443018193442/8440380939375\) | \(2033668868375612160000\) | \([2]\) | \(6291456\) | \(3.1826\) | |
| 64680.f2 | 64680bh4 | \([0, -1, 0, -23324016, -43348202820]\) | \(11647843478225136004/128410942275\) | \(15469997002456550400\) | \([2]\) | \(3145728\) | \(2.8360\) | |
| 64680.f3 | 64680bh3 | \([0, -1, 0, -5384136, 4099699836]\) | \(143279368983686884/22699269140625\) | \(2734639426688400000000\) | \([2, 2]\) | \(3145728\) | \(2.8360\) | |
| 64680.f4 | 64680bh2 | \([0, -1, 0, -1494516, -640969020]\) | \(12257375872392016/1191317675625\) | \(35880277304219040000\) | \([2, 2]\) | \(1572864\) | \(2.4895\) | |
| 64680.f5 | 64680bh1 | \([0, -1, 0, 112929, -48143304]\) | \(84611246065664/580054565475\) | \(-1091885433177092400\) | \([2]\) | \(786432\) | \(2.1429\) | \(\Gamma_0(N)\)-optimal |
| 64680.f6 | 64680bh5 | \([0, -1, 0, 9556944, 22788002700]\) | \(400647648358480318/1163177490234375\) | \(-280261977187500000000000\) | \([2]\) | \(6291456\) | \(3.1826\) |
Rank
sage: E.rank()
The elliptic curves in class 64680.f have rank \(1\).
Complex multiplication
The elliptic curves in class 64680.f do not have complex multiplication.Modular form 64680.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
