↑

Properties

Label 268770.a
Number of curves $2$
Conductor $268770$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 268770.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
268770.a1 268770a2 \([1, 1, 0, -63131333, 193044033357]\) \(1152829477932246539641/3188367360\) \(76959437149347840\) \([2]\) \(21565440\) \(2.8983\)  
268770.a2 268770a1 \([1, 1, 0, -3944133, 3017609037]\) \(-281115640967896441/468084326400\) \(-11298417726298521600\) \([2]\) \(10782720\) \(2.5518\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 268770.a have rank \(1\).

Complex multiplication

The elliptic curves in class 268770.a do not have complex multiplication.

Modular form 268770.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 4 q^{7} - q^{8} + q^{9} + q^{10} - 2 q^{11} - q^{12} + 2 q^{13} + 4 q^{14} + q^{15} + q^{16} - q^{18} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.