↑

Properties

Label 257754r
Number of curves $2$
Conductor $257754$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 257754r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
257754.r2 257754r1 \([1, 1, 0, 49056535834, 54541328363400852]\) \(2129503377881546170534943/210835998001488447189504\) \(-1292649473276724888738086705165320704\) \([]\) \(5132626560\) \(5.6086\) \(\Gamma_0(N)\)-optimal
257754.r1 257754r2 \([1, 1, 0, -9493146128126, 11260432875231620892]\) \(-15431857370630972204702226136417/3779323070318626304351304\) \(-23171280153759405752526111898134522504\) \([]\) \(15397879680\) \(6.1579\)  

Rank

sage: E.rank()
 

The elliptic curves in class 257754r have rank \(1\).

Complex multiplication

The elliptic curves in class 257754r do not have complex multiplication.

Modular form 257754.2.a.r

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + 3 q^{5} + q^{6} + q^{7} - q^{8} + q^{9} - 3 q^{10} - q^{12} + 4 q^{13} - q^{14} - 3 q^{15} + q^{16} - q^{17} - q^{18} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.