Properties

Label 443760k
Number of curves $2$
Conductor $443760$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 443760k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
443760.k2 443760k1 \([0, -1, 0, -185516, -3230340]\) \(235984/135\) \(403944201593890560\) \([]\) \(6241536\) \(2.0681\) \(\Gamma_0(N)\)-optimal*
443760.k1 443760k2 \([0, -1, 0, -9726356, 11678574156]\) \(34008433744/375\) \(1122067226649696000\) \([]\) \(18724608\) \(2.6174\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 443760k1.

Rank

sage: E.rank()
 

The elliptic curves in class 443760k have rank \(0\).

Complex multiplication

The elliptic curves in class 443760k do not have complex multiplication.

Modular form 443760.2.a.k

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{7} + q^{9} + 3 q^{11} - 4 q^{13} + q^{15} + 6 q^{17} - 8 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.