Properties

Label 443760.df
Number of curves $4$
Conductor $443760$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("df1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 443760.df

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
443760.df1 443760df3 \([0, 1, 0, -24806800, 47223347348]\) \(65202655558249/512820150\) \(13278094733281842585600\) \([2]\) \(34062336\) \(3.0732\) \(\Gamma_0(N)\)-optimal*
443760.df2 443760df2 \([0, 1, 0, -2618800, -409851052]\) \(76711450249/41602500\) \(1077184537583708160000\) \([2, 2]\) \(17031168\) \(2.7267\) \(\Gamma_0(N)\)-optimal*
443760.df3 443760df1 \([0, 1, 0, -2027120, -1110163500]\) \(35578826569/51600\) \(1336042837313126400\) \([2]\) \(8515584\) \(2.3801\) \(\Gamma_0(N)\)-optimal*
443760.df4 443760df4 \([0, 1, 0, 10102320, -3213585900]\) \(4403686064471/2721093750\) \(-70455383998934400000000\) \([4]\) \(34062336\) \(3.0732\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 443760.df1.

Rank

sage: E.rank()
 

The elliptic curves in class 443760.df have rank \(1\).

Complex multiplication

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

Modular form 443760.2.a.df

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{5} + 4q^{7} + q^{9} - 2q^{13} + q^{15} + 2q^{17} + 4q^{19} + O(q^{20})\)  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.