Properties

Label 198744x
Number of curves $4$
Conductor $198744$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("x1")
 
E.isogeny_class()
 

Elliptic curves in class 198744x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
198744.di4 198744x1 \([0, 1, 0, -60727, -37391782]\) \(-2725888/64827\) \(-589012160032990512\) \([2]\) \(2211840\) \(2.0905\) \(\Gamma_0(N)\)-optimal
198744.di3 198744x2 \([0, 1, 0, -2089572, -1158125760]\) \(6940769488/35721\) \(5192923533352079616\) \([2, 2]\) \(4423680\) \(2.4370\)  
198744.di2 198744x3 \([0, 1, 0, -3248912, 269253648]\) \(6522128932/3720087\) \(2163223574750666308608\) \([2]\) \(8847360\) \(2.7836\)  
198744.di1 198744x4 \([0, 1, 0, -33391752, -74280018240]\) \(7080974546692/189\) \(109903143563006976\) \([2]\) \(8847360\) \(2.7836\)  

Rank

sage: E.rank()
 

The elliptic curves in class 198744x have rank \(0\).

Complex multiplication

The elliptic curves in class 198744x do not have complex multiplication.

Modular form 198744.2.a.x

sage: E.q_eigenform(10)
 
\(q + q^{3} + 2 q^{5} + q^{9} + 2 q^{15} + 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.