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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("d1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

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

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(5\)\(1 + T\)
\(13\)\(1 - T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - 2 T + 3 T^{2}\) 1.3.ac
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(11\) \( 1 - 6 T + 11 T^{2}\) 1.11.ag
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 37570.2.a.d

Copy content 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

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 37570.d

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
37570.d1 37570b3 \([1, 1, 0, -59973, -5451187]\) \(988345570681/44994560\) \(1086059296624640\) \([2]\) \(331776\) \(1.6471\)  
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.d4 37570b4 \([1, 1, 0, 32507, -20636403]\) \(157376536199/7722894400\) \(-186411896459713600\) \([2]\) \(663552\) \(1.9936\)