↑

Properties

Label 330.e
Number of curves $6$
Conductor $330$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("e1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 330.e have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(11\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 330.e do not have complex multiplication.

Modular form 330.2.a.e

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{8} + q^{9} + q^{10} - q^{11} + q^{12} - 2 q^{13} + q^{15} + q^{16} + 2 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 330.e

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
330.e1 330b5 \([1, 0, 0, -3885, -93525]\) \(6484907238722641/283593750\) \(283593750\) \([2]\) \(256\) \(0.70009\)  
330.e2 330b4 \([1, 0, 0, -1175, 15405]\) \(179415687049201/1443420\) \(1443420\) \([4]\) \(128\) \(0.35352\)  
330.e3 330b3 \([1, 0, 0, -255, -1323]\) \(1834216913521/329422500\) \(329422500\) \([2, 2]\) \(128\) \(0.35352\)  
330.e4 330b2 \([1, 0, 0, -75, 225]\) \(46694890801/3920400\) \(3920400\) \([2, 4]\) \(64\) \(0.0069415\)  
330.e5 330b1 \([1, 0, 0, 5, 17]\) \(13651919/126720\) \(-126720\) \([4]\) \(32\) \(-0.33963\) \(\Gamma_0(N)\)-optimal
330.e6 330b6 \([1, 0, 0, 495, -7473]\) \(13411719834479/32153832150\) \(-32153832150\) \([2]\) \(256\) \(0.70009\)