Properties

Label 80586.q
Number of curves $4$
Conductor $80586$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("q1")
 
E.isogeny_class()
 

Elliptic curves in class 80586.q

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
80586.q1 80586p4 \([1, -1, 0, -6106487481, 183670297731949]\) \(19499096390516434897995817/15393430272\) \(19880122129322957568\) \([2]\) \(58982400\) \(3.9076\)  
80586.q2 80586p2 \([1, -1, 0, -381658041, 2869879323757]\) \(4760617885089919932457/133756441657344\) \(172742160047877049614336\) \([2, 2]\) \(29491200\) \(3.5610\)  
80586.q3 80586p3 \([1, -1, 0, -366324921, 3111023301997]\) \(-4209586785160189454377/801182513521564416\) \(-1034701553536009834034191104\) \([2]\) \(58982400\) \(3.9076\)  
80586.q4 80586p1 \([1, -1, 0, -24814521, 41038003309]\) \(1308451928740468777/194033737531392\) \(250588356927145899982848\) \([2]\) \(14745600\) \(3.2144\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 80586.q have rank \(0\).

Complex multiplication

The elliptic curves in class 80586.q do not have complex multiplication.

Modular form 80586.2.a.q

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + 2 q^{5} + 4 q^{7} - q^{8} - 2 q^{10} - 6 q^{13} - 4 q^{14} + q^{16} - 2 q^{17} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.