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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 54390.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
54390.i1 | 54390i4 | \([1, 1, 0, -513888, 63078912]\) | \(127568139540190201/59114336463360\) | \(6954742570577840640\) | \([2]\) | \(1741824\) | \(2.3101\) | |
54390.i2 | 54390i2 | \([1, 1, 0, -260313, -51226083]\) | \(16581570075765001/998001000\) | \(117413819649000\) | \([2]\) | \(580608\) | \(1.7608\) | |
54390.i3 | 54390i1 | \([1, 1, 0, -15313, -903083]\) | \(-3375675045001/999000000\) | \(-117531351000000\) | \([2]\) | \(290304\) | \(1.4142\) | \(\Gamma_0(N)\)-optimal |
54390.i4 | 54390i3 | \([1, 1, 0, 113312, 7508992]\) | \(1367594037332999/995878502400\) | \(-117164109928857600\) | \([2]\) | \(870912\) | \(1.9635\) |
Rank
sage: E.rank()
The elliptic curves in class 54390.i have rank \(1\).
Complex multiplication
The elliptic curves in class 54390.i do not have complex multiplication.Modular form 54390.2.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.