↑

Properties

Label 330330bg
Number of curves $6$
Conductor $330330$
CM no
Rank $2$
Graph

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 0, -45377, -3736971]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -45377, -3736971]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -45377, -3736971]) 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 330330bg have rank \(2\).

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.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 330330.2.a.bg

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);
 
\(q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + q^{7} - q^{8} + q^{9} - q^{10} - q^{12} - q^{13} - q^{14} - q^{15} + q^{16} - 2 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

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 330330bg

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
330330.bg5 330330bg1 \([1, 1, 0, -45377, -3736971]\) \(5832972054001/4542720\) \(8047705585920\) \([2]\) \(1310720\) \(1.4077\) \(\Gamma_0(N)\)-optimal
330330.bg4 330330bg2 \([1, 1, 0, -55057, -2039099]\) \(10418796526321/5038160400\) \(8925408476384400\) \([2, 2]\) \(2621440\) \(1.7542\)  
330330.bg2 330330bg3 \([1, 1, 0, -464037, 120082329]\) \(6237734630203441/82168222500\) \(145566018420322500\) \([2, 2]\) \(5242880\) \(2.1008\)  
330330.bg6 330330bg4 \([1, 1, 0, 199043, -15303119]\) \(492271755328079/342606902820\) \(-606949027366702020\) \([2]\) \(5242880\) \(2.1008\)  
330330.bg1 330330bg5 \([1, 1, 0, -7400967, 7746543171]\) \(25306558948218234961/4478906250\) \(7934655635156250\) \([2]\) \(10485760\) \(2.4474\)  
330330.bg3 330330bg6 \([1, 1, 0, -70787, 317572479]\) \(-22143063655441/24584858584650\) \(-43553576659081138650\) \([2]\) \(10485760\) \(2.4474\)