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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 26640.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
26640.x1 | 26640bk4 | \([0, 0, 0, -1510203, -318864022]\) | \(127568139540190201/59114336463360\) | \(176514462850209546240\) | \([2]\) | \(1161216\) | \(2.5796\) | |
26640.x2 | 26640bk2 | \([0, 0, 0, -765003, 257525498]\) | \(16581570075765001/998001000\) | \(2980015017984000\) | \([2]\) | \(387072\) | \(2.0303\) | |
26640.x3 | 26640bk1 | \([0, 0, 0, -45003, 4517498]\) | \(-3375675045001/999000000\) | \(-2982998016000000\) | \([2]\) | \(193536\) | \(1.6837\) | \(\Gamma_0(N)\)-optimal |
26640.x4 | 26640bk3 | \([0, 0, 0, 332997, -37591702]\) | \(1367594037332999/995878502400\) | \(-2973677274110361600\) | \([2]\) | \(580608\) | \(2.2330\) |
Rank
sage: E.rank()
The elliptic curves in class 26640.x have rank \(0\).
Complex multiplication
The elliptic curves in class 26640.x do not have complex multiplication.Modular form 26640.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.