Properties

Label 825.a
Number of curves 44
Conductor 825825
CM no
Rank 11
Graph

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Copy content sage:E = EllipticCurve("a1") E.isogeny_class()
 

Elliptic curves in class 825.a

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
825.a1 825b3 [1,0,0,3663,84942][1, 0, 0, -3663, 84942] 347873904937/395307347873904937/395307 61766718756176671875 [2][2] 768768 0.792090.79209  
825.a2 825b2 [1,0,0,288,567][1, 0, 0, -288, 567] 169112377/88209169112377/88209 13782656251378265625 [2,2][2, 2] 384384 0.445510.44551  
825.a3 825b1 [1,0,0,163,808][1, 0, 0, -163, -808] 30664297/29730664297/297 46406254640625 [2][2] 192192 0.0989400.098940 Γ0(N)\Gamma_0(N)-optimal
825.a4 825b4 [1,0,0,1087,4692][1, 0, 0, 1087, 4692] 9090072503/58458519090072503/5845851 91341421875-91341421875 [2][2] 768768 0.792090.79209  

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 825.a have rank 11.

L-function data

 
Bad L-factors:
Prime L-Factor
331T1 - T
5511
11111T1 - T
 
Good L-factors:
Prime L-Factor Isogeny Class over Fp\mathbb{F}_p
22 1+T+2T2 1 + T + 2 T^{2} 1.2.b
77 1+4T+7T2 1 + 4 T + 7 T^{2} 1.7.e
1313 12T+13T2 1 - 2 T + 13 T^{2} 1.13.ac
1717 12T+17T2 1 - 2 T + 17 T^{2} 1.17.ac
1919 1+19T2 1 + 19 T^{2} 1.19.a
2323 1+8T+23T2 1 + 8 T + 23 T^{2} 1.23.i
2929 1+6T+29T2 1 + 6 T + 29 T^{2} 1.29.g
\cdots\cdots\cdots
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 825.a do not have complex multiplication.

Modular form 825.2.a.a

Copy content sage:E.q_eigenform(10)
 
qq2+q3q4q64q7+3q8+q9+q11q12+2q13+4q14q16+2q17q18+O(q20)q - q^{2} + q^{3} - q^{4} - q^{6} - 4 q^{7} + 3 q^{8} + q^{9} + q^{11} - q^{12} + 2 q^{13} + 4 q^{14} - q^{16} + 2 q^{17} - q^{18} + O(q^{20}) Copy content Toggle raw display

Isogeny matrix

Copy content sage:E.isogeny_class().matrix()
 

The i,ji,j entry is the smallest degree of a cyclic isogeny between the ii-th and jj-th curve in the isogeny class, in the LMFDB numbering.

(1244212242144241)\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 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.