Properties

Label 288990gr
Number of curves $4$
Conductor $288990$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("gr1")
 
E.isogeny_class()
 

Elliptic curves in class 288990gr

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
288990.gr4 288990gr1 \([1, -1, 1, 14198503, 43783417721]\) \(89962967236397039/287450726400000\) \(-1011465450114917990400000\) \([2]\) \(41472000\) \(3.2893\) \(\Gamma_0(N)\)-optimal
288990.gr3 288990gr2 \([1, -1, 1, -133764377, 513062487929]\) \(75224183150104868881/11219310000000000\) \(39477877065224910000000000\) \([2]\) \(82944000\) \(3.6359\)  
288990.gr2 288990gr3 \([1, -1, 1, -5021528297, 136963897027481]\) \(-3979640234041473454886161/1471455901872240\) \(-5177676274299572719094640\) \([2]\) \(207360000\) \(4.0940\)  
288990.gr1 288990gr4 \([1, -1, 1, -80344459877, 8765627517933329]\) \(16300610738133468173382620881/2228489100\) \(7841482117081505100\) \([2]\) \(414720000\) \(4.4406\)  

Rank

sage: E.rank()
 

The elliptic curves in class 288990gr have rank \(1\).

Complex multiplication

The elliptic curves in class 288990gr do not have complex multiplication.

Modular form 288990.2.a.gr

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{5} + 2 q^{7} + q^{8} + q^{10} + 2 q^{11} + 2 q^{14} + q^{16} + 2 q^{17} + q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.