Show commands:
SageMath
E = EllipticCurve("dm1")
E.isogeny_class()
Elliptic curves in class 115920dm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 115920.cb4 | 115920dm1 | \([0, 0, 0, -109443, 13933442]\) | \(48551226272641/9273600\) | \(27690821222400\) | \([2]\) | \(393216\) | \(1.5802\) | \(\Gamma_0(N)\)-optimal |
| 115920.cb3 | 115920dm2 | \([0, 0, 0, -120963, 10820738]\) | \(65553197996161/20996010000\) | \(62693749923840000\) | \([2, 2]\) | \(786432\) | \(1.9267\) | |
| 115920.cb6 | 115920dm3 | \([0, 0, 0, 342717, 73788482]\) | \(1490881681033919/1650501562500\) | \(-4928371257600000000\) | \([2]\) | \(1572864\) | \(2.2733\) | |
| 115920.cb2 | 115920dm4 | \([0, 0, 0, -768963, -251360062]\) | \(16840406336564161/604708416900\) | \(1805649657528729600\) | \([2, 2]\) | \(1572864\) | \(2.2733\) | |
| 115920.cb5 | 115920dm5 | \([0, 0, 0, 289437, -889998622]\) | \(898045580910239/115117148363070\) | \(-343737963137753210880\) | \([2]\) | \(3145728\) | \(2.6199\) | |
| 115920.cb1 | 115920dm6 | \([0, 0, 0, -12195363, -16392292702]\) | \(67176973097223766561/91487391870\) | \(273179888325550080\) | \([2]\) | \(3145728\) | \(2.6199\) |
Rank
sage: E.rank()
The elliptic curves in class 115920dm have rank \(0\).
Complex multiplication
The elliptic curves in class 115920dm do not have complex multiplication.Modular form 115920.2.a.dm
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
