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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 271040.x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 271040.x1 | 271040x4 | \([0, 1, 0, -27259041, -54489137441]\) | \(4823468134087681/30382271150\) | \(14109650887839619481600\) | \([2]\) | \(26542080\) | \(3.0875\) | |
| 271040.x2 | 271040x2 | \([0, 1, 0, -2091041, 1113169759]\) | \(2177286259681/105875000\) | \(49168782688256000000\) | \([2]\) | \(8847360\) | \(2.5382\) | |
| 271040.x3 | 271040x3 | \([0, 1, 0, -697121, -1848724385]\) | \(-80677568161/3131816380\) | \(-1454428325929072721920\) | \([2]\) | \(13271040\) | \(2.7409\) | |
| 271040.x4 | 271040x1 | \([0, 1, 0, 77279, 67605855]\) | \(109902239/4312000\) | \(-2002510422212608000\) | \([2]\) | \(4423680\) | \(2.1916\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 271040.x have rank \(2\).
Complex multiplication
The elliptic curves in class 271040.x do not have complex multiplication.Modular form 271040.2.a.x
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
