Properties

Label 21168dr
Number of curves $3$
Conductor $21168$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("dr1")
 
E.isogeny_class()
 

Elliptic curves in class 21168dr

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
21168.r2 21168dr1 \([0, 0, 0, -2499, 48706]\) \(-132651/2\) \(-26022076416\) \([]\) \(18144\) \(0.80159\) \(\Gamma_0(N)\)-optimal
21168.r3 21168dr2 \([0, 0, 0, 9261, 240786]\) \(9261/8\) \(-75880374829056\) \([]\) \(54432\) \(1.3509\)  
21168.r1 21168dr3 \([0, 0, 0, -96579, -15317694]\) \(-1167051/512\) \(-43707095901536256\) \([]\) \(163296\) \(1.9002\)  

Rank

sage: E.rank()
 

The elliptic curves in class 21168dr have rank \(0\).

Complex multiplication

The elliptic curves in class 21168dr do not have complex multiplication.

Modular form 21168.2.a.dr

sage: E.q_eigenform(10)
 
\(q - 3 q^{5} + 3 q^{11} + 4 q^{13} + 2 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.