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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ee1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 57330ee 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 \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 4 T + 23 T^{2}\) 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 57330ee do not have complex multiplication.

Modular form 57330.2.a.ee

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - q^{5} + q^{8} - q^{10} - 6 q^{11} - q^{13} + q^{16} - 8 q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels.

Elliptic curves in class 57330ee

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
57330.db3 57330ee1 \([1, -1, 1, -334508, -33353589]\) \(48264326765929/22299191460\) \(1912515152960526660\) \([2]\) \(1327104\) \(2.2026\) \(\Gamma_0(N)\)-optimal
57330.db4 57330ee2 \([1, -1, 1, 1178122, -252382413]\) \(2108526614950391/1540302022350\) \(-132105729625414804350\) \([2]\) \(2654208\) \(2.5492\)  
57330.db1 57330ee3 \([1, -1, 1, -22699823, -41621948553]\) \(15082569606665230489/7751016000\) \(664774576128936000\) \([2]\) \(3981312\) \(2.7519\)  
57330.db2 57330ee4 \([1, -1, 1, -22576343, -42097247769]\) \(-14837772556740428569/342100087875000\) \(-29340597530797882875000\) \([2]\) \(7962624\) \(3.0985\)