Show commands:
SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 115920.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 115920.d1 | 115920q6 | \([0, 0, 0, -12799443, -17625225742]\) | \(155324313723954725282/13018359375\) | \(19436306400000000\) | \([2]\) | \(3670016\) | \(2.5680\) | |
| 115920.d2 | 115920q4 | \([0, 0, 0, -1101603, 444555218]\) | \(198048499826486404/242568272835\) | \(181076245398236160\) | \([2]\) | \(1835008\) | \(2.2214\) | |
| 115920.d3 | 115920q3 | \([0, 0, 0, -801723, -274123078]\) | \(76343005935514084/694180580625\) | \(518203026714240000\) | \([2, 2]\) | \(1835008\) | \(2.2214\) | |
| 115920.d4 | 115920q5 | \([0, 0, 0, -234723, -654353278]\) | \(-957928673903042/123339801817575\) | \(-184145337395224934400\) | \([2]\) | \(3670016\) | \(2.5680\) | |
| 115920.d5 | 115920q2 | \([0, 0, 0, -87303, 2928998]\) | \(394315384276816/208332909225\) | \(38879920851206400\) | \([2, 2]\) | \(917504\) | \(1.8748\) | |
| 115920.d6 | 115920q1 | \([0, 0, 0, 20742, 357527]\) | \(84611246065664/53699121315\) | \(-626346551018160\) | \([2]\) | \(458752\) | \(1.5282\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 115920.d have rank \(0\).
Complex multiplication
The elliptic curves in class 115920.d do not have complex multiplication.Modular form 115920.2.a.d
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.
