Properties

Label 33600.gj
Number of curves $4$
Conductor $33600$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("gj1")
 
E.isogeny_class()
 

Elliptic curves in class 33600.gj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
33600.gj1 33600cq4 \([0, 1, 0, -1412033, -646295937]\) \(608119035935048/826875\) \(423360000000000\) \([2]\) \(294912\) \(2.0808\)  
33600.gj2 33600cq3 \([0, 1, 0, -224033, 27192063]\) \(2428799546888/778248135\) \(398463045120000000\) \([2]\) \(294912\) \(2.0808\)  
33600.gj3 33600cq2 \([0, 1, 0, -89033, -9932937]\) \(1219555693504/43758225\) \(2800526400000000\) \([2, 2]\) \(147456\) \(1.7343\)  
33600.gj4 33600cq1 \([0, 1, 0, 2092, -547062]\) \(1012048064/130203045\) \(-130203045000000\) \([2]\) \(73728\) \(1.3877\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 33600.gj have rank \(1\).

Complex multiplication

The elliptic curves in class 33600.gj do not have complex multiplication.

Modular form 33600.2.a.gj

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{7} + q^{9} + 2 q^{13} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.