Show commands:
SageMath
E = EllipticCurve("cv1")
E.isogeny_class()
Elliptic curves in class 19110.cv
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19110.cv1 | 19110cn5 | \([1, 0, 0, -441736, 112966790]\) | \(81025909800741361/11088090\) | \(1304502700410\) | \([2]\) | \(196608\) | \(1.7370\) | |
| 19110.cv2 | 19110cn3 | \([1, 0, 0, -41406, -3243864]\) | \(66730743078481/60937500\) | \(7169235937500\) | \([2]\) | \(98304\) | \(1.3905\) | |
| 19110.cv3 | 19110cn4 | \([1, 0, 0, -27686, 1752960]\) | \(19948814692561/231344100\) | \(27217402020900\) | \([2, 2]\) | \(98304\) | \(1.3905\) | |
| 19110.cv4 | 19110cn6 | \([1, 0, 0, -5636, 4473930]\) | \(-168288035761/73415764890\) | \(-8637291323543610\) | \([2]\) | \(196608\) | \(1.7370\) | |
| 19110.cv5 | 19110cn2 | \([1, 0, 0, -3186, -25740]\) | \(30400540561/15210000\) | \(1789441290000\) | \([2, 2]\) | \(49152\) | \(1.0439\) | |
| 19110.cv6 | 19110cn1 | \([1, 0, 0, 734, -3004]\) | \(371694959/249600\) | \(-29365190400\) | \([2]\) | \(24576\) | \(0.69731\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 19110.cv have rank \(0\).
Complex multiplication
The elliptic curves in class 19110.cv do not have complex multiplication.Modular form 19110.2.a.cv
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
