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Properties

Label 471510.ep
Number of curves $6$
Conductor $471510$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ep1")
 
E.isogeny_class()
 

Elliptic curves in class 471510.ep

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
471510.ep1 471510ep5 \([1, -1, 1, -467737952, 3893723161319]\) \(3216206300355197383681/57660\) \(202890765259260\) \([2]\) \(62914560\) \(3.2136\) \(\Gamma_0(N)\)-optimal*
471510.ep2 471510ep3 \([1, -1, 1, -29233652, 60844775879]\) \(785209010066844481/3324675600\) \(11698681524848931600\) \([2, 2]\) \(31457280\) \(2.8670\) \(\Gamma_0(N)\)-optimal*
471510.ep3 471510ep6 \([1, -1, 1, -28777352, 62835704039]\) \(-749011598724977281/51173462246460\) \(-180066301008500169336060\) \([2]\) \(62914560\) \(3.2136\)  
471510.ep4 471510ep4 \([1, -1, 1, -5627732, -4005018169]\) \(5601911201812801/1271193750000\) \(4473005076834693750000\) \([2]\) \(31457280\) \(2.8670\)  
471510.ep5 471510ep2 \([1, -1, 1, -1855652, 919809479]\) \(200828550012481/12454560000\) \(43824405296000160000\) \([2, 2]\) \(15728640\) \(2.5205\) \(\Gamma_0(N)\)-optimal*
471510.ep6 471510ep1 \([1, -1, 1, 91228, 60067271]\) \(23862997439/457113600\) \(-1608465628068249600\) \([2]\) \(7864320\) \(2.1739\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 471510.ep1.

Rank

sage: E.rank()
 

The elliptic curves in class 471510.ep have rank \(1\).

Complex multiplication

The elliptic curves in class 471510.ep do not have complex multiplication.

Modular form 471510.2.a.ep

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{5} + q^{8} + q^{10} - 4 q^{11} + q^{16} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.