Properties

Label 8112.s
Number of curves 44
Conductor 81128112
CM no
Rank 11
Graph

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Show commands: SageMath
E = EllipticCurve("s1")
 
E.isogeny_class()
 

Elliptic curves in class 8112.s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8112.s1 8112bh4 [0,1,0,187984,31307732][0, 1, 0, -187984, 31307732] 37159393753/105337159393753/1053 2081845197619220818451976192 [4][4] 4300843008 1.65731.6573  
8112.s2 8112bh3 [0,1,0,52784,4244460][0, 1, 0, -52784, -4244460] 822656953/85683822656953/85683 16940051478405121694005147840512 [2][2] 4300843008 1.65731.6573  
8112.s3 8112bh2 [0,1,0,12224,444276][0, 1, 0, -12224, 444276] 10218313/152110218313/1521 3007109729894430071097298944 [2,2][2, 2] 2150421504 1.31071.3107  
8112.s4 8112bh1 [0,1,0,1296,38676][0, 1, 0, 1296, 38676] 12167/3912167/39 771053776896-771053776896 [2][2] 1075210752 0.964110.96411 Γ0(N)\Gamma_0(N)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 8112.s have rank 11.

Complex multiplication

The elliptic curves in class 8112.s do not have complex multiplication.

Modular form 8112.2.a.s

sage: E.q_eigenform(10)
 
q+q32q54q7+q9+4q112q15+2q17+O(q20)q + q^{3} - 2 q^{5} - 4 q^{7} + q^{9} + 4 q^{11} - 2 q^{15} + 2 q^{17} + O(q^{20}) Copy content Toggle raw display

Isogeny matrix

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.

(1424412422124421)\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)

Isogeny graph

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

The vertices are labelled with LMFDB labels.