Show commands:
SageMath
E = EllipticCurve("dp1")
E.isogeny_class()
Elliptic curves in class 29040.dp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 29040.dp1 | 29040bp4 | \([0, 1, 0, -1277800, -556384732]\) | \(127191074376964/495\) | \(897968839680\) | \([2]\) | \(245760\) | \(1.9298\) | |
| 29040.dp2 | 29040bp2 | \([0, 1, 0, -79900, -8704852]\) | \(124386546256/245025\) | \(111123643910400\) | \([2, 2]\) | \(122880\) | \(1.5832\) | |
| 29040.dp3 | 29040bp3 | \([0, 1, 0, -53280, -14571900]\) | \(-9220796644/45106875\) | \(-81827410515840000\) | \([4]\) | \(245760\) | \(1.9298\) | |
| 29040.dp4 | 29040bp1 | \([0, 1, 0, -6695, -37380]\) | \(1171019776/658845\) | \(18674945712720\) | \([2]\) | \(61440\) | \(1.2366\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 29040.dp have rank \(0\).
Complex multiplication
The elliptic curves in class 29040.dp do not have complex multiplication.Modular form 29040.2.a.dp
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
