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Properties

Label 25992.w
Number of curves $6$
Conductor $25992$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("w1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 25992.w have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(19\)\(1\)
 
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.e
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 25992.w do not have complex multiplication.

Modular form 25992.2.a.w

Copy content sage:E.q_eigenform(10)
 
\(q + 2 q^{5} - 4 q^{11} + 2 q^{13} - 2 q^{17} + O(q^{20})\) 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 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 25992.w

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
25992.w1 25992bc6 \([0, 0, 0, -1248699, -537073418]\) \(3065617154/9\) \(632152115693568\) \([2]\) \(221184\) \(2.0693\)  
25992.w2 25992bc4 \([0, 0, 0, -209019, 36777958]\) \(28756228/3\) \(105358685948928\) \([2]\) \(110592\) \(1.7227\)  
25992.w3 25992bc3 \([0, 0, 0, -79059, -8162210]\) \(1556068/81\) \(2844684520621056\) \([2, 2]\) \(110592\) \(1.7227\)  
25992.w4 25992bc2 \([0, 0, 0, -14079, 480130]\) \(35152/9\) \(79019014461696\) \([2, 2]\) \(55296\) \(1.3762\)  
25992.w5 25992bc1 \([0, 0, 0, 2166, 48013]\) \(2048/3\) \(-1646229467952\) \([2]\) \(27648\) \(1.0296\) \(\Gamma_0(N)\)-optimal
25992.w6 25992bc5 \([0, 0, 0, 50901, -32360762]\) \(207646/6561\) \(-460838892340611072\) \([2]\) \(221184\) \(2.0693\)