Properties

Label 289800cf
Number of curves $6$
Conductor $289800$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
sage: E = EllipticCurve("cf1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 289800cf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
289800.cf6 289800cf1 \([0, 0, 0, 518550, -44690875]\) \(84611246065664/53699121315\) \(-9786664859658750000\) \([2]\) \(5505024\) \(2.3330\) \(\Gamma_0(N)\)-optimal
289800.cf5 289800cf2 \([0, 0, 0, -2182575, -366124750]\) \(394315384276816/208332909225\) \(607498763300100000000\) \([2, 2]\) \(11010048\) \(2.6795\)  
289800.cf3 289800cf3 \([0, 0, 0, -20043075, 34265384750]\) \(76343005935514084/694180580625\) \(8096922292410000000000\) \([2, 2]\) \(22020096\) \(3.0261\)  
289800.cf2 289800cf4 \([0, 0, 0, -27540075, -55569402250]\) \(198048499826486404/242568272835\) \(2829316334347440000000\) \([2]\) \(22020096\) \(3.0261\)  
289800.cf1 289800cf5 \([0, 0, 0, -319986075, 2203153217750]\) \(155324313723954725282/13018359375\) \(303692287500000000000\) \([2]\) \(44040192\) \(3.3727\)  
289800.cf4 289800cf6 \([0, 0, 0, -5868075, 81794159750]\) \(-957928673903042/123339801817575\) \(-2877270896800389600000000\) \([2]\) \(44040192\) \(3.3727\)  

Rank

sage: E.rank()
 

The elliptic curves in class 289800cf have rank \(1\).

Complex multiplication

The elliptic curves in class 289800cf do not have complex multiplication.

Modular form 289800.2.a.cf

sage: E.q_eigenform(10)
 
\(q - q^{7} + 4q^{11} + 2q^{13} + 2q^{17} - 4q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.