Properties

Label 278850ey
Number of curves $8$
Conductor $278850$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("ey1")
 
E.isogeny_class()
 

Elliptic curves in class 278850ey

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
278850.ey7 278850ey1 \([1, 1, 1, -48249588, -123419094219]\) \(164711681450297281/8097103872000\) \(610674591301632000000000\) \([2]\) \(55738368\) \(3.3234\) \(\Gamma_0(N)\)-optimal
278850.ey6 278850ey2 \([1, 1, 1, -134777588, 442820137781]\) \(3590017885052913601/954068544000000\) \(71954791168689000000000000\) \([2, 2]\) \(111476736\) \(3.6700\)  
278850.ey3 278850ey3 \([1, 1, 1, -3860889588, -92339337174219]\) \(84392862605474684114881/11228954880\) \(846875319927780000000\) \([2]\) \(167215104\) \(3.8727\)  
278850.ey5 278850ey4 \([1, 1, 1, -1993777588, 34261748137781]\) \(11621808143080380273601/1335706803288000\) \(100737525304246062375000000\) \([2]\) \(222953472\) \(4.0165\)  
278850.ey8 278850ey5 \([1, 1, 1, 339774412, 2864933545781]\) \(57519563401957999679/80296734375000000\) \(-6055890627372802734375000000\) \([2]\) \(222953472\) \(4.0165\)  
278850.ey2 278850ey6 \([1, 1, 1, -3861227588, -92322361462219]\) \(84415028961834287121601/30783551683856400\) \(2321661317493800409806250000\) \([2, 2]\) \(334430208\) \(4.2193\)  
278850.ey1 278850ey7 \([1, 1, 1, -4423575088, -63669631642219]\) \(126929854754212758768001/50235797102795981820\) \(3788728087155462034572068437500\) \([2]\) \(668860416\) \(4.5658\)  
278850.ey4 278850ey8 \([1, 1, 1, -3304288088, -119888638954219]\) \(-52902632853833942200321/51713453577420277500\) \(-3900171299196474878429648437500\) \([2]\) \(668860416\) \(4.5658\)  

Rank

sage: E.rank()
 

The elliptic curves in class 278850ey have rank \(1\).

Complex multiplication

The elliptic curves in class 278850ey do not have complex multiplication.

Modular form 278850.2.a.ey

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} - 4 q^{7} + q^{8} + q^{9} + q^{11} - q^{12} - 4 q^{14} + q^{16} + 6 q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.