Properties

Label 5712t
Number of curves $6$
Conductor $5712$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("5712.o1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 5712t

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5712.o5 5712t1 [0, 1, 0, -1123904, -458409420] [2] 92160 \(\Gamma_0(N)\)-optimal
5712.o4 5712t2 [0, 1, 0, -1451584, -169657804] [2, 2] 184320  
5712.o2 5712t3 [0, 1, 0, -13744704, 19474747956] [2, 4] 368640  
5712.o6 5712t4 [0, 1, 0, 5598656, -1328717260] [2] 368640  
5712.o1 5712t5 [0, 1, 0, -219497664, 1251605773620] [8] 737280  
5712.o3 5712t6 [0, 1, 0, -4681664, 44782380852] [4] 737280  

Rank

sage: E.rank()
 

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

Modular form 5712.2.a.o

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

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.