Properties

Label 5712bb
Number of curves $4$
Conductor $5712$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("bb1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 5712bb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5712.q4 5712bb1 \([0, 1, 0, 16, -6444]\) \(103823/4386816\) \(-17968398336\) \([2]\) \(4608\) \(0.64655\) \(\Gamma_0(N)\)-optimal
5712.q3 5712bb2 \([0, 1, 0, -5104, -139564]\) \(3590714269297/73410624\) \(300689915904\) \([2, 2]\) \(9216\) \(0.99312\)  
5712.q1 5712bb3 \([0, 1, 0, -81264, -8943660]\) \(14489843500598257/6246072\) \(25583910912\) \([2]\) \(18432\) \(1.3397\)  
5712.q2 5712bb4 \([0, 1, 0, -10864, 226772]\) \(34623662831857/14438442312\) \(59139859709952\) \([4]\) \(18432\) \(1.3397\)  

Rank

sage: E.rank()
 

The elliptic curves in class 5712bb have rank \(1\).

Complex multiplication

The elliptic curves in class 5712bb do not have complex multiplication.

Modular form 5712.2.a.bb

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2q^{5} + q^{7} + q^{9} - 6q^{13} - 2q^{15} + q^{17} + O(q^{20})\)  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 Cremona 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 Cremona labels.