Properties

Label 12675e
Number of curves 88
Conductor 1267512675
CM no
Rank 11
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 12675e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12675.j6 12675e1 [1,1,1,464838,121785906][1, 1, 1, -464838, 121785906] 147281603041/5265147281603041/5265 397080459140625397080459140625 [4][4] 9676896768 1.89031.8903 Γ0(N)\Gamma_0(N)-optimal
12675.j5 12675e2 [1,1,1,485963,110082656][1, 1, 1, -485963, 110082656] 168288035761/27720225168288035761/27720225 20906286173753906252090628617375390625 [2,2][2, 2] 193536193536 2.23692.2369  
12675.j4 12675e3 [1,1,1,2197088,1149305344][1, 1, 1, -2197088, -1149305344] 15551989015681/144590062515551989015681/1445900625 109048221091494140625109048221091494140625 [2,2][2, 2] 387072387072 2.58352.5835  
12675.j7 12675e4 [1,1,1,887162,620885156][1, 1, 1, 887162, 620885156] 1023887723039/27980368651023887723039/2798036865 211024836286152890625-211024836286152890625 [2][2] 387072387072 2.58352.5835  
12675.j2 12675e5 [1,1,1,34328213,77428596094][1, 1, 1, -34328213, -77428596094] 59319456301170001/59414062559319456301170001/594140625 4480942681274414062544809426812744140625 [2,2][2, 2] 774144774144 2.93002.9300  
12675.j8 12675e6 [1,1,1,2556037,5436624094][1, 1, 1, 2556037, -5436624094] 24487529386319/18353941222524487529386319/183539412225 13842338855974062890625-13842338855974062890625 [2][2] 774144774144 2.93002.9300  
12675.j1 12675e7 [1,1,1,549250088,4954768596094][1, 1, 1, -549250088, -4954768596094] 242970740812818720001/24375242970740812818720001/24375 18383354589843751838335458984375 [2][2] 15482881548288 3.27663.2766  
12675.j3 12675e8 [1,1,1,33504338,81320581594][1, 1, 1, -33504338, -81320581594] 55150149867714721/5950927734375-55150149867714721/5950927734375 448812367916107177734375-448812367916107177734375 [2][2] 15482881548288 3.27663.2766  

Rank

sage: E.rank()
 

The elliptic curves in class 12675e have rank 11.

Complex multiplication

The elliptic curves in class 12675e do not have complex multiplication.

Modular form 12675.2.a.e

sage: E.q_eigenform(10)
 
qq2q3q4+q6+3q8+q94q11+q12q162q17q18+4q19+O(q20)q - q^{2} - q^{3} - q^{4} + q^{6} + 3 q^{8} + q^{9} - 4 q^{11} + q^{12} - q^{16} - 2 q^{17} - q^{18} + 4 q^{19} + 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 Cremona numbering.

(124488161621224488421422444241881616842814228428418816841628141684162841)\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)

Isogeny graph

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

The vertices are labelled with Cremona labels.