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Properties

Label 303240.d
Number of curves $4$
Conductor $303240$
CM no
Rank $1$
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 303240.d have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 303240.d do not have complex multiplication.

Modular form 303240.2.a.d

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - q^{5} - q^{7} + q^{9} - 6 q^{13} + q^{15} - 2 q^{17} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 303240.d

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
303240.d1 303240d4 \([0, -1, 0, -73144496, -233729369604]\) \(898353183174324196/29899176238575\) \(1440392281413659709004800\) \([2]\) \(53084160\) \(3.4085\)  
303240.d2 303240d2 \([0, -1, 0, -11232996, 9434237796]\) \(13015144447800784/4341909875625\) \(52292857682272908960000\) \([2, 2]\) \(26542080\) \(3.0619\)  
303240.d3 303240d1 \([0, -1, 0, -10104871, 12364655296]\) \(151591373397612544/32558203125\) \(24507669596681250000\) \([2]\) \(13271040\) \(2.7153\) \(\Gamma_0(N)\)-optimal
303240.d4 303240d3 \([0, -1, 0, 32628504, 65033075196]\) \(79743193254623804/84085819746075\) \(-4050832864830765739084800\) \([2]\) \(53084160\) \(3.4085\)