Properties

Label 32487.f
Number of curves $6$
Conductor $32487$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 32487.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
32487.f1 32487l6 \([1, 0, 0, -988527, 378154692]\) \(908031902324522977/161726530797\) \(19026964621736253\) \([2]\) \(393216\) \(2.1281\)  
32487.f2 32487l4 \([1, 0, 0, -68062, 4629995]\) \(296380748763217/92608836489\) \(10895337004094361\) \([2, 2]\) \(196608\) \(1.7815\)  
32487.f3 32487l2 \([1, 0, 0, -26657, -1622160]\) \(17806161424897/668584449\) \(78658291840401\) \([2, 2]\) \(98304\) \(1.4350\)  
32487.f4 32487l1 \([1, 0, 0, -26412, -1654353]\) \(17319700013617/25857\) \(3042050193\) \([2]\) \(49152\) \(1.0884\) \(\Gamma_0(N)\)-optimal
32487.f5 32487l3 \([1, 0, 0, 10828, -5812983]\) \(1193377118543/124806800313\) \(-14683395250024137\) \([2]\) \(196608\) \(1.7815\)  
32487.f6 32487l5 \([1, 0, 0, 189923, 31408838]\) \(6439735268725823/7345472585373\) \(-864187504196548077\) \([2]\) \(393216\) \(2.1281\)  

Rank

sage: E.rank()
 

The elliptic curves in class 32487.f have rank \(1\).

Complex multiplication

The elliptic curves in class 32487.f do not have complex multiplication.

Modular form 32487.2.a.f

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} - q^{4} + 2 q^{5} - q^{6} + 3 q^{8} + q^{9} - 2 q^{10} + 4 q^{11} - q^{12} - q^{13} + 2 q^{15} - q^{16} - q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.