Properties

Label 479370.t
Number of curves $4$
Conductor $479370$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("479370.t1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 479370.t

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
479370.t1 479370t3 [1, 1, 0, -2556657, 1572399219] [2] 9289728 \(\Gamma_0(N)\)-optimal*
479370.t2 479370t4 [1, 1, 0, -185037, 16222911] [2] 9289728  
479370.t3 479370t2 [1, 1, 0, -159807, 24513489] [2, 2] 4644864 \(\Gamma_0(N)\)-optimal*
479370.t4 479370t1 [1, 1, 0, -8427, 504621] [2] 2322432 \(\Gamma_0(N)\)-optimal*
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 479370.t4.

Rank

sage: E.rank()
 

The elliptic curves in class 479370.t have rank \(1\).

Modular form 479370.2.a.t

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + 4q^{7} - q^{8} + q^{9} - q^{10} + 4q^{11} - q^{12} - 2q^{13} - 4q^{14} - q^{15} + q^{16} + 2q^{17} - q^{18} + q^{19} + O(q^{20}) \)

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.