↑

Properties

Label 57330.ep
Number of curves $4$
Conductor $57330$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("ep1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 57330.ep have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(7\)\(1\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 57330.ep do not have complex multiplication.

Modular form 57330.2.a.ep

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 57330.ep

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
57330.ep1 57330dh4 \([1, -1, 1, -1763642, 901936459]\) \(261984288445803/42250\) \(97837702530750\) \([2]\) \(995328\) \(2.0868\)  
57330.ep2 57330dh3 \([1, -1, 1, -109892, 14203459]\) \(-63378025803/812500\) \(-1881494279437500\) \([2]\) \(497664\) \(1.7402\)  
57330.ep3 57330dh2 \([1, -1, 1, -24632, 897411]\) \(520300455507/193072360\) \(613298792204280\) \([2]\) \(331776\) \(1.5375\)  
57330.ep4 57330dh1 \([1, -1, 1, 4768, 97731]\) \(3774555693/3515200\) \(-11166113649600\) \([2]\) \(165888\) \(1.1909\) \(\Gamma_0(N)\)-optimal