Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("z1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 75712z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
75712.bw4 75712z1 [0, 0, 0, 676, 35152] [2] 73728 \(\Gamma_0(N)\)-optimal
75712.bw3 75712z2 [0, 0, 0, -12844, 527280] [2, 2] 147456  
75712.bw2 75712z3 [0, 0, 0, -39884, -2425488] [2] 294912  
75712.bw1 75712z4 [0, 0, 0, -202124, 34976240] [2] 294912  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 75712.2.a.z

sage: E.q_eigenform(10)
 
\( q + 2q^{5} + q^{7} - 3q^{9} - 4q^{11} - 6q^{17} + 8q^{19} + O(q^{20}) \)

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.