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Properties

Label 17238.e
Number of curves $6$
Conductor $17238$
CM no
Rank $1$
Graph

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

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 17238.e have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 - T\)
\(13\)\(1\)
\(17\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 17238.e do not have complex multiplication.

Modular form 17238.2.a.e

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 17238.e

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
17238.e1 17238f5 \([1, 0, 1, -4688740, -3908190232]\) \(2361739090258884097/5202\) \(25109060418\) \([2]\) \(294912\) \(2.1296\)  
17238.e2 17238f3 \([1, 0, 1, -293050, -61082344]\) \(576615941610337/27060804\) \(130617332294436\) \([2, 2]\) \(147456\) \(1.7830\)  
17238.e3 17238f6 \([1, 0, 1, -277840, -67701736]\) \(-491411892194497/125563633938\) \(-606071678364643842\) \([2]\) \(294912\) \(2.1296\)  
17238.e4 17238f2 \([1, 0, 1, -19270, -850744]\) \(163936758817/30338064\) \(146436040357776\) \([2, 2]\) \(73728\) \(1.4364\)  
17238.e5 17238f1 \([1, 0, 1, -5750, 155144]\) \(4354703137/352512\) \(1701508094208\) \([2]\) \(36864\) \(1.0898\) \(\Gamma_0(N)\)-optimal
17238.e6 17238f4 \([1, 0, 1, 38190, -4941896]\) \(1276229915423/2927177028\) \(-14128924423343652\) \([2]\) \(147456\) \(1.7830\)