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Properties

Label 486720.gb
Number of curves $4$
Conductor $486720$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("gb1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 486720.gb have rank \(1\).

Complex multiplication

The elliptic curves in class 486720.gb do not have complex multiplication.

Modular form 486720.2.a.gb

Copy content sage:E.q_eigenform(10)
 
\(q - q^{5} + 2 q^{7} - 2 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 486720.gb

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
486720.gb1 486720gb4 \([0, 0, 0, -84940186188, -9528290341377392]\) \(73474353581350183614361/576510977802240\) \(531783852042391779232878428160\) \([2]\) \(1114767360\) \(4.8769\)  
486720.gb2 486720gb3 \([0, 0, 0, -5195981388, -155507383686512]\) \(-16818951115904497561/1592332281446400\) \(-1468795264900377098560575897600\) \([2]\) \(557383680\) \(4.5303\)  
486720.gb3 486720gb2 \([0, 0, 0, -1557749388, 890845500688]\) \(453198971846635561/261896250564000\) \(241577701592627342528741376000\) \([2]\) \(371589120\) \(4.3276\) \(\Gamma_0(N)\)-optimal*
486720.gb4 486720gb1 \([0, 0, 0, 389130612, 111314748688]\) \(7064514799444439/4094064000000\) \(-3776436543719919845376000000\) \([2]\) \(185794560\) \(3.9810\) \(\Gamma_0(N)\)-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 486720.gb1.