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Properties

Label 332350.bi
Number of curves $2$
Conductor $332350$
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, 0, 1, -8099376, -9185271852]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, -8099376, -9185271852]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, -8099376, -9185271852]) 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 332350.bi have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(5\)\(1\)
\(17\)\(1\)
\(23\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - T + 3 T^{2}\) 1.3.ab
\(7\) \( 1 + T + 7 T^{2}\) 1.7.b
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(19\) \( 1 - 5 T + 19 T^{2}\) 1.19.af
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 332350.bi do not have complex multiplication.

Modular form 332350.2.a.bi

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);
 
\(q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} - q^{8} - 2 q^{9} + q^{12} + 4 q^{13} + q^{14} + q^{16} + 2 q^{18} + 5 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

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 332350.bi

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
332350.bi1 332350bi2 \([1, 0, 1, -8099376, -9185271852]\) \(-155799034954129/6463718750\) \(-2437788395698730468750\) \([]\) \(17915904\) \(2.8708\)  
332350.bi2 332350bi1 \([1, 0, 1, 498374, -37265852]\) \(36297569231/22599800\) \(-8523503623221875000\) \([]\) \(5971968\) \(2.3215\) \(\Gamma_0(N)\)-optimal