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Properties

Label 103488gu
Number of curves $4$
Conductor $103488$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("gu1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 103488gu have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(7\)\(1\)
\(11\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 - 2 T + 23 T^{2}\) 1.23.ac
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 103488gu do not have complex multiplication.

Modular form 103488.2.a.gu

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

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 103488gu

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
103488.g3 103488gu1 \([0, -1, 0, -141185, 14930721]\) \(10091699281/2737152\) \(84416448599949312\) \([2]\) \(1382400\) \(1.9558\) \(\Gamma_0(N)\)-optimal
103488.g4 103488gu2 \([0, -1, 0, 360575, 96717601]\) \(168105213359/228637728\) \(-7051411472114515968\) \([2]\) \(2764800\) \(2.3024\)  
103488.g1 103488gu3 \([0, -1, 0, -31563905, -68244436959]\) \(112763292123580561/1932612\) \(59603646988419072\) \([2]\) \(6912000\) \(2.7605\)  
103488.g2 103488gu4 \([0, -1, 0, -31532545, -68386842719]\) \(-112427521449300721/466873642818\) \(-14398840426697819947008\) \([2]\) \(13824000\) \(3.1071\)