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Properties

Label 330330.ec
Number of curves $4$
Conductor $330330$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ec1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 330330.ec have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 + T\)
\(5\)\(1 + T\)
\(7\)\(1 - T\)
\(11\)\(1\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 8 T + 19 T^{2}\) 1.19.ai
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 330330.ec do not have complex multiplication.

Modular form 330330.2.a.ec

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{7} + q^{8} + q^{9} - q^{10} - q^{12} - q^{13} + q^{14} + q^{15} + q^{16} + 6 q^{17} + q^{18} + 8 q^{19} + O(q^{20})\) 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 330330.ec

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
330330.ec1 330330ec4 \([1, 1, 1, -19085882186, -1014892106087617]\) \(434014578033107719741685694649/103121648659575000\) \(182686291021005346575000\) \([2]\) \(589824000\) \(4.2821\)  
330330.ec2 330330ec2 \([1, 1, 1, -1193007186, -15854166187617]\) \(105997782562506306791694649/51649016225625000000\) \(91499382833684450625000000\) \([2, 2]\) \(294912000\) \(3.9355\)  
330330.ec3 330330ec3 \([1, 1, 1, -994242906, -21310007156481]\) \(-61354313914516350666047929/75227254486083984375000\) \(-133269670184621429443359375000\) \([2]\) \(589824000\) \(4.2821\)  
330330.ec4 330330ec1 \([1, 1, 1, -87125266, -158605273441]\) \(41285728533151645510969/17760741842188800000\) \(31464237578689832716800000\) \([2]\) \(147456000\) \(3.5889\) \(\Gamma_0(N)\)-optimal