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Properties

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

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

Rank

Copy content sage:E.rank()
 

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

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(13\)\(1\)
\(19\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 2 T + 7 T^{2}\) 1.7.ac
\(11\) \( 1 - 2 T + 11 T^{2}\) 1.11.ac
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 288990.2.a.gr

Copy content 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

Copy content 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 LMFDB 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

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 288990.gr

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