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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 4026.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4026.i1 | 4026i4 | \([1, 0, 0, -42603, 159705]\) | \(8551551109433208625/4937300515763352\) | \(4937300515763352\) | \([2]\) | \(31968\) | \(1.7008\) | |
| 4026.i2 | 4026i2 | \([1, 0, 0, -30003, 1997793]\) | \(2986886106831048625/15277413888\) | \(15277413888\) | \([6]\) | \(10656\) | \(1.1515\) | |
| 4026.i3 | 4026i1 | \([1, 0, 0, -1843, 32225]\) | \(-692332063944625/52241891328\) | \(-52241891328\) | \([6]\) | \(5328\) | \(0.80494\) | \(\Gamma_0(N)\)-optimal |
| 4026.i4 | 4026i3 | \([1, 0, 0, 10637, 21281]\) | \(133100178546359375/77205251969472\) | \(-77205251969472\) | \([2]\) | \(15984\) | \(1.3542\) |
Rank
sage: E.rank()
The elliptic curves in class 4026.i have rank \(0\).
Complex multiplication
The elliptic curves in class 4026.i do not have complex multiplication.Modular form 4026.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
