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Properties

Label 25410.w
Number of curves $8$
Conductor $25410$
CM no
Rank $1$
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 25410.w have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 - T\)
\(5\)\(1 + T\)
\(7\)\(1 + T\)
\(11\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 25410.w do not have complex multiplication.

Modular form 25410.2.a.w

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{7} - q^{8} + q^{9} + q^{10} + q^{12} - 2 q^{13} + q^{14} - q^{15} + q^{16} + 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 25410.w

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
25410.w1 25410v8 \([1, 0, 1, -780574, -167063734]\) \(29689921233686449/10380965400750\) \(18390513446318070750\) \([2]\) \(829440\) \(2.3980\)  
25410.w2 25410v5 \([1, 0, 1, -697084, -224072158]\) \(21145699168383889/2593080\) \(4593799397880\) \([2]\) \(276480\) \(1.8487\)  
25410.w3 25410v6 \([1, 0, 1, -326824, 69975266]\) \(2179252305146449/66177562500\) \(117237588800062500\) \([2, 2]\) \(414720\) \(2.0514\)  
25410.w4 25410v3 \([1, 0, 1, -324404, 71090402]\) \(2131200347946769/2058000\) \(3645872538000\) \([2]\) \(207360\) \(1.7048\)  
25410.w5 25410v2 \([1, 0, 1, -43684, -3484318]\) \(5203798902289/57153600\) \(101251088769600\) \([2, 2]\) \(138240\) \(1.5021\)  
25410.w6 25410v4 \([1, 0, 1, -9804, -8742494]\) \(-58818484369/18600435000\) \(-32951805229035000\) \([2]\) \(276480\) \(1.8487\)  
25410.w7 25410v1 \([1, 0, 1, -4964, 46946]\) \(7633736209/3870720\) \(6857216593920\) \([2]\) \(69120\) \(1.1555\) \(\Gamma_0(N)\)-optimal
25410.w8 25410v7 \([1, 0, 1, 88206, 235655242]\) \(42841933504271/13565917968750\) \(-24032851202636718750\) \([2]\) \(829440\) \(2.3980\)