Properties

Label 75712.bx
Number of curves $4$
Conductor $75712$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("bx1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 75712.bx

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
75712.bx1 75712y4 \([0, 0, 0, -639589964, 6223771098480]\) \(22868021811807457713/8953460393696\) \(11328983717494232465801216\) \([2]\) \(23224320\) \(3.7722\)  
75712.bx2 75712y3 \([0, 0, 0, -338472524, -2350626226832]\) \(3389174547561866673/74853681183008\) \(94713786405296186712915968\) \([2]\) \(23224320\) \(3.7722\)  
75712.bx3 75712y2 \([0, 0, 0, -46007884, 65950600560]\) \(8511781274893233/3440817243136\) \(4353731496908956113043456\) \([2, 2]\) \(11612160\) \(3.4256\)  
75712.bx4 75712y1 \([0, 0, 0, 9370036, 7493668208]\) \(71903073502287/60782804992\) \(-76909639153911209132032\) \([2]\) \(5806080\) \(3.0791\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 75712.bx have rank \(0\).

Complex multiplication

The elliptic curves in class 75712.bx do not have complex multiplication.

Modular form 75712.2.a.bx

sage: E.q_eigenform(10)
 
\(q + 2q^{5} + q^{7} - 3q^{9} + 4q^{11} - 6q^{17} + O(q^{20})\)  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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.