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Properties

Label 451770.c
Number of curves $6$
Conductor $451770$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("c1")
 
E.isogeny_class()
 

Elliptic curves in class 451770.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
451770.c1 451770c6 \([1, 1, 0, -234215393, -1379754999687]\) \(553808571467029327441/12529687500\) \(32147750115267187500\) \([2]\) \(77856768\) \(3.2672\)  
451770.c2 451770c3 \([1, 1, 0, -16188453, 25008455133]\) \(182864522286982801/463015182960\) \(1187970282688438790640\) \([2]\) \(38928384\) \(2.9206\) \(\Gamma_0(N)\)-optimal*
451770.c3 451770c4 \([1, 1, 0, -14655173, -21511566723]\) \(135670761487282321/643043610000\) \(1649873972315696490000\) \([2, 2]\) \(38928384\) \(2.9206\)  
451770.c4 451770c5 \([1, 1, 0, -7125673, -43577519423]\) \(-15595206456730321/310672490129100\) \(-797100612474023689401900\) \([2]\) \(77856768\) \(3.2672\)  
451770.c5 451770c2 \([1, 1, 0, -1403253, 59908653]\) \(119102750067601/68309049600\) \(175262332532410886400\) \([2, 2]\) \(19464192\) \(2.5741\) \(\Gamma_0(N)\)-optimal*
451770.c6 451770c1 \([1, 1, 0, 349067, 7689517]\) \(1833318007919/1070530560\) \(-2746688529433559040\) \([2]\) \(9732096\) \(2.2275\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 451770.c1.

Rank

sage: E.rank()
 

The elliptic curves in class 451770.c have rank \(2\).

Complex multiplication

The elliptic curves in class 451770.c do not have complex multiplication.

Modular form 451770.2.a.c

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.