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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 31878.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 31878.i1 | 31878m4 | \([1, -1, 0, -62028, 5961384]\) | \(36204575259448513/1527466248\) | \(1113522894792\) | \([2]\) | \(110592\) | \(1.3916\) | |
| 31878.i2 | 31878m2 | \([1, -1, 0, -4068, 84240]\) | \(10214075575873/1806590016\) | \(1317004121664\) | \([2, 2]\) | \(55296\) | \(1.0451\) | |
| 31878.i3 | 31878m1 | \([1, -1, 0, -1188, -14256]\) | \(254478514753/21762048\) | \(15864532992\) | \([2]\) | \(27648\) | \(0.69850\) | \(\Gamma_0(N)\)-optimal |
| 31878.i4 | 31878m3 | \([1, -1, 0, 7812, 476280]\) | \(72318867421247/177381135624\) | \(-129310847869896\) | \([2]\) | \(110592\) | \(1.3916\) |
Rank
sage: E.rank()
The elliptic curves in class 31878.i have rank \(1\).
Complex multiplication
The elliptic curves in class 31878.i do not have complex multiplication.Modular form 31878.2.a.i
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.
