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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("gd1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 330330.gd 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 \(\mathbb{F}_p\)
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 8 T + 19 T^{2}\) 1.19.i
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 330330.2.a.gd

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

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
330330.gd1 330330gd3 \([1, 0, 0, -40044948006, 1917291845078820]\) \(4008766897254067912673785886329/1423480510711669921875000000\) \(2521782557036876678466796875000000\) \([2]\) \(2202009600\) \(5.1102\)  
330330.gd2 330330gd2 \([1, 0, 0, -16927133286, -825826437529884]\) \(302773487204995438715379645049/8911747415025000000000000\) \(15787704162309104025000000000000\) \([2, 2]\) \(1101004800\) \(4.7637\)  
330330.gd3 330330gd1 \([1, 0, 0, -16805707366, -838559135215900]\) \(296304326013275547793071733369/268420373544960000000\) \(475523065377682882560000000\) \([2]\) \(550502400\) \(4.4171\) \(\Gamma_0(N)\)-optimal
330330.gd4 330330gd4 \([1, 0, 0, 4247866714, -2754051582529884]\) \(4784981304203817469820354951/1852343836482910078035000000\) \(-3281540099303500660753762635000000\) \([2]\) \(2202009600\) \(5.1102\)