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SageMath
E = EllipticCurve("dp1")
E.isogeny_class()
Elliptic curves in class 90090.dp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 90090.dp1 | 90090dr4 | \([1, -1, 1, -710873627, 7293494514251]\) | \(54497099771831721530744218729/16209843781074944000000\) | \(11816976116403634176000000\) | \([6]\) | \(34836480\) | \(3.7894\) | |
| 90090.dp2 | 90090dr3 | \([1, -1, 1, -50270747, 82089235019]\) | \(19272683606216463573689449/7161126378530668544000\) | \(5220461129948857368576000\) | \([6]\) | \(17418240\) | \(3.4428\) | |
| 90090.dp3 | 90090dr2 | \([1, -1, 1, -23698292, -31447740841]\) | \(2019051077229077416165369/582160888682835862400\) | \(424395287849787343689600\) | \([2]\) | \(11612160\) | \(3.2401\) | |
| 90090.dp4 | 90090dr1 | \([1, -1, 1, -21722612, -38958485929]\) | \(1555006827939811751684089/221961497899581440\) | \(161809931968794869760\) | \([2]\) | \(5806080\) | \(2.8935\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 90090.dp have rank \(0\).
Complex multiplication
The elliptic curves in class 90090.dp do not have complex multiplication.Modular form 90090.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
