Properties

Label 288990.ej
Number of curves $4$
Conductor $288990$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("ej1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 288990.ej

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
288990.ej1 288990ej3 \([1, -1, 1, -944573, 353069331]\) \(26487576322129/44531250\) \(156694058107031250\) \([2]\) \(4718592\) \(2.1947\)  
288990.ej2 288990ej2 \([1, -1, 1, -77603, 1773087]\) \(14688124849/8122500\) \(28580996198722500\) \([2, 2]\) \(2359296\) \(1.8481\)  
288990.ej3 288990ej1 \([1, -1, 1, -47183, -3909369]\) \(3301293169/22800\) \(80227357750800\) \([2]\) \(1179648\) \(1.5016\) \(\Gamma_0(N)\)-optimal
288990.ej4 288990ej4 \([1, -1, 1, 302647, 13788987]\) \(871257511151/527800050\) \(-1857193132992988050\) \([2]\) \(4718592\) \(2.1947\)  

Rank

sage: E.rank()
 

The elliptic curves in class 288990.ej have rank \(0\).

Complex multiplication

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

Modular form 288990.2.a.ej

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