Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("bw1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 75712.bw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
75712.bw1 75712z4 \([0, 0, 0, -202124, 34976240]\) \(1443468546/7\) \(4428616564736\) \([2]\) \(294912\) \(1.6270\)  
75712.bw2 75712z3 \([0, 0, 0, -39884, -2425488]\) \(11090466/2401\) \(1519015481704448\) \([2]\) \(294912\) \(1.6270\)  
75712.bw3 75712z2 \([0, 0, 0, -12844, 527280]\) \(740772/49\) \(15500157976576\) \([2, 2]\) \(147456\) \(1.2804\)  
75712.bw4 75712z1 \([0, 0, 0, 676, 35152]\) \(432/7\) \(-553577070592\) \([2]\) \(73728\) \(0.93387\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 75712.2.a.bw

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