Properties

Label 37570b
Number of curves $4$
Conductor $37570$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 37570b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
37570.d2 37570b1 \([1, 1, 0, -9398, 344708]\) \(3803721481/26000\) \(627576794000\) \([2]\) \(110592\) \(1.0977\) \(\Gamma_0(N)\)-optimal
37570.d3 37570b2 \([1, 1, 0, -3618, 771272]\) \(-217081801/10562500\) \(-254953072562500\) \([2]\) \(221184\) \(1.4443\)  
37570.d1 37570b3 \([1, 1, 0, -59973, -5451187]\) \(988345570681/44994560\) \(1086059296624640\) \([2]\) \(331776\) \(1.6471\)  
37570.d4 37570b4 \([1, 1, 0, 32507, -20636403]\) \(157376536199/7722894400\) \(-186411896459713600\) \([2]\) \(663552\) \(1.9936\)  

Rank

sage: E.rank()
 

The elliptic curves in class 37570b have rank \(0\).

Complex multiplication

The elliptic curves in class 37570b do not have complex multiplication.

Modular form 37570.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} + 2 q^{3} + q^{4} - q^{5} - 2 q^{6} + 4 q^{7} - q^{8} + q^{9} + q^{10} + 6 q^{11} + 2 q^{12} + q^{13} - 4 q^{14} - 2 q^{15} + q^{16} - q^{18} + 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.