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Properties

Label 39360.bh
Number of curves $4$
Conductor $39360$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 0, -335905, 75045025]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -335905, 75045025]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -335905, 75045025]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 39360.bh have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(5\)\(1 - T\)
\(41\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 39360.bh do not have complex multiplication.

Modular form 39360.2.a.bh

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q - q^{3} + q^{5} + q^{9} + 4 q^{11} - 2 q^{13} - q^{15} - 6 q^{17} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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

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

The vertices are labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 39360.bh

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
39360.bh1 39360r4 \([0, -1, 0, -335905, 75045025]\) \(15989485458638089/615000\) \(161218560000\) \([4]\) \(147456\) \(1.6414\)  
39360.bh2 39360r3 \([0, -1, 0, -33825, -410463]\) \(16327137318409/9155465640\) \(2400050384732160\) \([2]\) \(147456\) \(1.6414\)  
39360.bh3 39360r2 \([0, -1, 0, -21025, 1174177]\) \(3921141001609/24206400\) \(6345562521600\) \([2, 2]\) \(73728\) \(1.2948\)  
39360.bh4 39360r1 \([0, -1, 0, -545, 39585]\) \(-68417929/2519040\) \(-660351221760\) \([2]\) \(36864\) \(0.94822\) \(\Gamma_0(N)\)-optimal