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Properties

Label 244758e
Number of curves $2$
Conductor $244758$
CM no
Rank $0$
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, 0, -599628, -180198576]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -599628, -180198576]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -599628, -180198576]) 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 244758e have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(19\)\(1\)
\(113\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + T + 5 T^{2}\) 1.5.b
\(7\) \( 1 - T + 7 T^{2}\) 1.7.ab
\(11\) \( 1 + 2 T + 11 T^{2}\) 1.11.c
\(13\) \( 1 + 7 T + 13 T^{2}\) 1.13.h
\(17\) \( 1 + 3 T + 17 T^{2}\) 1.17.d
\(23\) \( 1 - 3 T + 23 T^{2}\) 1.23.ad
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 244758e do not have complex multiplication.

Modular form 244758.2.a.e

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);
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + q^{7} - q^{8} + q^{9} + q^{10} - 2 q^{11} - q^{12} - 7 q^{13} - q^{14} + q^{15} + q^{16} - 3 q^{17} - q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

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 244758e

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
244758.e2 244758e1 \([1, 1, 0, -599628, -180198576]\) \(-506814405937489/4048994304\) \(-190488504195661824\) \([]\) \(3302208\) \(2.1437\) \(\Gamma_0(N)\)-optimal
244758.e1 244758e2 \([1, 1, 0, -2570688, 17604675804]\) \(-39934705050538129/2823126576537804\) \(-132816476967734918985324\) \([]\) \(23115456\) \(3.1167\)