Properties

Label 5712j
Number of curves $4$
Conductor $5712$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("j1")
 
E.isogeny_class()
 

Elliptic curves in class 5712j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5712.x4 5712j1 \([0, 1, 0, 28, 60]\) \(9148592/9639\) \(-2467584\) \([2]\) \(768\) \(-0.086725\) \(\Gamma_0(N)\)-optimal
5712.x3 5712j2 \([0, 1, 0, -152, 420]\) \(381775972/127449\) \(130507776\) \([2, 2]\) \(1536\) \(0.25985\)  
5712.x2 5712j3 \([0, 1, 0, -992, -12012]\) \(52767497666/1753941\) \(3592071168\) \([2]\) \(3072\) \(0.60642\)  
5712.x1 5712j4 \([0, 1, 0, -2192, 38772]\) \(569001644066/122451\) \(250779648\) \([2]\) \(3072\) \(0.60642\)  

Rank

sage: E.rank()
 

The elliptic curves in class 5712j have rank \(0\).

Complex multiplication

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

Modular form 5712.2.a.j

sage: E.q_eigenform(10)
 
\(q + q^{3} + 2 q^{5} - q^{7} + q^{9} + 4 q^{11} + 2 q^{13} + 2 q^{15} - q^{17} + 4 q^{19} + O(q^{20})\) 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 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.