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SageMath
E = EllipticCurve("du1")
E.isogeny_class()
Elliptic curves in class 103488.du
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
103488.du1 | 103488gh4 | \([0, -1, 0, -126214657, 545816017825]\) | \(7209828390823479793/49509306\) | \(1526915489226817536\) | \([2]\) | \(7077888\) | \(3.0889\) | |
103488.du2 | 103488gh3 | \([0, -1, 0, -10998017, 1197723297]\) | \(4770223741048753/2740574865798\) | \(84522013133706075045888\) | \([2]\) | \(7077888\) | \(3.0889\) | |
103488.du3 | 103488gh2 | \([0, -1, 0, -7893377, 8519085345]\) | \(1763535241378513/4612311396\) | \(142248201008710680576\) | \([2, 2]\) | \(3538944\) | \(2.7423\) | |
103488.du4 | 103488gh1 | \([0, -1, 0, -304257, 236319777]\) | \(-100999381393/723148272\) | \(-22302601000393900032\) | \([2]\) | \(1769472\) | \(2.3958\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 103488.du have rank \(0\).
Complex multiplication
The elliptic curves in class 103488.du do not have complex multiplication.Modular form 103488.2.a.du
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.