Show commands:
SageMath
E = EllipticCurve("dr1")
E.isogeny_class()
Elliptic curves in class 141120dr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
141120.dm3 | 141120dr1 | \([0, 0, 0, -71148, -6920368]\) | \(1771561/105\) | \(2360722772459520\) | \([2]\) | \(786432\) | \(1.7029\) | \(\Gamma_0(N)\)-optimal |
141120.dm2 | 141120dr2 | \([0, 0, 0, -212268, 29037008]\) | \(47045881/11025\) | \(247875891108249600\) | \([2, 2]\) | \(1572864\) | \(2.0494\) | |
141120.dm1 | 141120dr3 | \([0, 0, 0, -3175788, 2178181712]\) | \(157551496201/13125\) | \(295090346557440000\) | \([2]\) | \(3145728\) | \(2.3960\) | |
141120.dm4 | 141120dr4 | \([0, 0, 0, 493332, 181164368]\) | \(590589719/972405\) | \(-21862653595747614720\) | \([2]\) | \(3145728\) | \(2.3960\) |
Rank
sage: E.rank()
The elliptic curves in class 141120dr have rank \(0\).
Complex multiplication
The elliptic curves in class 141120dr do not have complex multiplication.Modular form 141120.2.a.dr
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.