↑

Properties

Label 422142w
Number of curves $2$
Conductor $422142$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("w1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 422142w have rank \(1\).

Complex multiplication

The elliptic curves in class 422142w do not have complex multiplication.

Modular form 422142.2.a.w

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - 3 q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + 3 q^{10} - 3 q^{11} + q^{12} - 4 q^{13} - q^{14} - 3 q^{15} + q^{16} - 6 q^{17} - q^{18} + q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 422142w

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
422142.w2 422142w1 \([1, 0, 1, 2396620470, -119337464426180]\) \(5440141067713791011601894407/25133280338487142984974336\) \(-7033322303202580580058203160576\) \([3]\) \(1308026880\) \(4.6014\) \(\Gamma_0(N)\)-optimal
422142.w1 422142w2 \([1, 0, 1, -130833560985, -18238826504070788]\) \(-885054173129909473160896402567273/1345692322481263587272687616\) \(-376579885215479283525976175149056\) \([]\) \(3924080640\) \(5.1507\)