Show commands:
SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 1680.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1680.b1 | 1680k7 | \([0, -1, 0, -103216, -7995584]\) | \(29689921233686449/10380965400750\) | \(42520434281472000\) | \([2]\) | \(13824\) | \(1.8922\) | |
| 1680.b2 | 1680k4 | \([0, -1, 0, -92176, -10740800]\) | \(21145699168383889/2593080\) | \(10621255680\) | \([2]\) | \(4608\) | \(1.3429\) | |
| 1680.b3 | 1680k6 | \([0, -1, 0, -43216, 3380416]\) | \(2179252305146449/66177562500\) | \(271063296000000\) | \([2, 2]\) | \(6912\) | \(1.5456\) | |
| 1680.b4 | 1680k3 | \([0, -1, 0, -42896, 3433920]\) | \(2131200347946769/2058000\) | \(8429568000\) | \([2]\) | \(3456\) | \(1.1990\) | |
| 1680.b5 | 1680k2 | \([0, -1, 0, -5776, -165440]\) | \(5203798902289/57153600\) | \(234101145600\) | \([2, 2]\) | \(2304\) | \(0.99629\) | |
| 1680.b6 | 1680k5 | \([0, -1, 0, -1296, -419904]\) | \(-58818484369/18600435000\) | \(-76187381760000\) | \([2]\) | \(4608\) | \(1.3429\) | |
| 1680.b7 | 1680k1 | \([0, -1, 0, -656, 2496]\) | \(7633736209/3870720\) | \(15854469120\) | \([2]\) | \(1152\) | \(0.64972\) | \(\Gamma_0(N)\)-optimal |
| 1680.b8 | 1680k8 | \([0, -1, 0, 11664, 11327040]\) | \(42841933504271/13565917968750\) | \(-55566000000000000\) | \([2]\) | \(13824\) | \(1.8922\) |
Rank
sage: E.rank()
The elliptic curves in class 1680.b have rank \(0\).
Complex multiplication
The elliptic curves in class 1680.b do not have complex multiplication.Modular form 1680.2.a.b
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
