Properties

Label 21315.p
Number of curves 44
Conductor 2131521315
CM no
Rank 11
Graph

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

Elliptic curves in class 21315.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
21315.p1 21315c4 [1,1,0,12618,532323][1, 1, 0, -12618, 532323] 1888690601881/318276451888690601881/31827645 37444906066053744490606605 [2][2] 5529655296 1.21061.2106  
21315.p2 21315c2 [1,1,0,1593,12312][1, 1, 0, -1593, -12312] 3803721481/17030253803721481/1703025 200359188225200359188225 [2,2][2, 2] 2764827648 0.864030.86403  
21315.p3 21315c1 [1,1,0,1348,19613][1, 1, 0, -1348, -19613] 2305199161/13052305199161/1305 153531945153531945 [2][2] 1382413824 0.517460.51746 Γ0(N)\Gamma_0(N)-optimal
21315.p4 21315c3 [1,1,0,5512,84783][1, 1, 0, 5512, -84783] 157376536199/118918125157376536199/118918125 13990598488125-13990598488125 [2][2] 5529655296 1.21061.2106  

Rank

sage: E.rank()
 

The elliptic curves in class 21315.p have rank 11.

Complex multiplication

The elliptic curves in class 21315.p do not have complex multiplication.

Modular form 21315.2.a.p

sage: E.q_eigenform(10)
 
q+q2q3q4q5q63q8+q9q104q11+q126q13+q15q166q17+q18+4q19+O(q20)q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} - 3 q^{8} + q^{9} - q^{10} - 4 q^{11} + q^{12} - 6 q^{13} + q^{15} - q^{16} - 6 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 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

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

The vertices are labelled with LMFDB labels.