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Properties

Label 25230p
Number of curves $8$
Conductor $25230$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("p1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 25230p have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 + T\)
\(5\)\(1 + T\)
\(29\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 6 T + 19 T^{2}\) 1.19.g
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 25230p do not have complex multiplication.

Modular form 25230.2.a.p

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 25230p

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
25230.o8 25230p1 \([1, 1, 1, 1244, 52373]\) \(357911/2160\) \(-1284818373360\) \([2]\) \(48384\) \(1.0054\) \(\Gamma_0(N)\)-optimal
25230.o6 25230p2 \([1, 1, 1, -15576, 671349]\) \(702595369/72900\) \(43362620100900\) \([2, 2]\) \(96768\) \(1.3519\)  
25230.o7 25230p3 \([1, 1, 1, -11371, -1532071]\) \(-273359449/1536000\) \(-913648621056000\) \([2]\) \(145152\) \(1.5547\)  
25230.o5 25230p4 \([1, 1, 1, -57626, -4610131]\) \(35578826569/5314410\) \(3161135005355610\) \([2]\) \(193536\) \(1.6985\)  
25230.o4 25230p5 \([1, 1, 1, -242646, 45903693]\) \(2656166199049/33750\) \(20075287083750\) \([2]\) \(193536\) \(1.6985\)  
25230.o3 25230p6 \([1, 1, 1, -280491, -57186087]\) \(4102915888729/9000000\) \(5353409889000000\) \([2, 2]\) \(290304\) \(1.9012\)  
25230.o1 25230p7 \([1, 1, 1, -4485491, -3658348087]\) \(16778985534208729/81000\) \(48180689001000\) \([2]\) \(580608\) \(2.2478\)  
25230.o2 25230p8 \([1, 1, 1, -381411, -12498711]\) \(10316097499609/5859375000\) \(3485292896484375000\) \([2]\) \(580608\) \(2.2478\)