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Properties

Label 75690bs
Number of curves $4$
Conductor $75690$
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([1, -1, 1, -152582, 22978181]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, -152582, 22978181]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, -152582, 22978181]) 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 75690bs have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 75690bs do not have complex multiplication.

Modular form 75690.2.a.bs

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);
 
\(q + q^{2} + q^{4} + q^{5} - 2 q^{7} + q^{8} + q^{10} + 6 q^{13} - 2 q^{14} + q^{16} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 75690bs

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
75690.bo3 75690bs1 \([1, -1, 1, -152582, 22978181]\) \(22095784790981/450000\) \(8000811450000\) \([2]\) \(358400\) \(1.5957\) \(\Gamma_0(N)\)-optimal
75690.bo4 75690bs2 \([1, -1, 1, -147362, 24619349]\) \(-19904714311301/3164062500\) \(-56255705507812500\) \([2]\) \(716800\) \(1.9423\)  
75690.bo1 75690bs3 \([1, -1, 1, -1516307, -709369549]\) \(21685195471991381/309586821120\) \(5504323962635550720\) \([2]\) \(1792000\) \(2.4004\)  
75690.bo2 75690bs4 \([1, -1, 1, -179987, -1916868301]\) \(-36267977929301/89261680665600\) \(-1587035281590169113600\) \([2]\) \(3584000\) \(2.7470\)