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Properties

Label 333270.dj
Number of curves $2$
Conductor $333270$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("dj1")
 
E.isogeny_class()
 

Elliptic curves in class 333270.dj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.dj1 333270dj1 \([1, -1, 1, -80243, -7442493]\) \(14295828483/2254000\) \(9009168132762000\) \([2]\) \(2433024\) \(1.7842\) \(\Gamma_0(N)\)-optimal
333270.dj2 333270dj2 \([1, -1, 1, 141937, -41480469]\) \(79119341757/231437500\) \(-925048513631812500\) \([2]\) \(4866048\) \(2.1308\)  

Rank

sage: E.rank()
 

The elliptic curves in class 333270.dj have rank \(1\).

Complex multiplication

The elliptic curves in class 333270.dj do not have complex multiplication.

Modular form 333270.2.a.dj

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - q^{10} + 4 q^{11} - q^{14} + q^{16} + 6 q^{17} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.