↑

Properties

Label 42042k
Number of curves $4$
Conductor $42042$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 42042k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42042.g4 42042k1 \([1, 1, 0, -15141554950, -717146114477036]\) \(3263224124812796801735447265625/1837810787484672\) \(216216601336784176128\) \([2]\) \(39813120\) \(4.1298\) \(\Gamma_0(N)\)-optimal
42042.g3 42042k2 \([1, 1, 0, -15141641190, -717137536994892]\) \(3263279883032933444452132257625/77441472526453540753248\) \(9110911801264732616078873952\) \([2]\) \(79626240\) \(4.4763\)  
42042.g2 42042k3 \([1, 1, 0, -15171567205, -714160487661683]\) \(3282666836869681281754155591625/26942969374939856448258048\) \(3169813403992299171281111089152\) \([2]\) \(119439360\) \(4.6791\)  
42042.g1 42042k4 \([1, 1, 0, -25857048165, 422135694720909]\) \(16250708692977087048493451847625/8749977648266474863605153792\) \(1029426120340902501228282738475008\) \([2]\) \(238878720\) \(5.0256\)  

Rank

sage: E.rank()
 

The elliptic curves in class 42042k have rank \(0\).

Complex multiplication

The elliptic curves in class 42042k do not have complex multiplication.

Modular form 42042.2.a.k

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

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.