Properties

Label 463680.fh
Number of curves $8$
Conductor $463680$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("fh1")
 
E.isogeny_class()
 

Elliptic curves in class 463680.fh

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
463680.fh1 463680fh8 \([0, 0, 0, -146245833228, 20197303934266352]\) \(1810117493172631097464564372609/125368453502655029296875000\) \(23958284560875000000000000000000000\) \([2]\) \(3057647616\) \(5.3448\)  
463680.fh2 463680fh6 \([0, 0, 0, -143722884108, 20971765552195568]\) \(1718043013877225552292911401729/9180538178765625000000\) \(1754428167243730944000000000000\) \([2, 2]\) \(1528823808\) \(4.9982\)  
463680.fh3 463680fh3 \([0, 0, 0, -143722699788, 20971822033225712]\) \(1718036403880129446396978632449/49057344000000\) \(9375004433055744000000\) \([2]\) \(764411904\) \(4.6516\) \(\Gamma_0(N)\)-optimal*
463680.fh4 463680fh7 \([0, 0, 0, -141202884108, 21742612384195568]\) \(-1629247127728109256861881401729/125809119536174660320875000\) \(-24042497151302717243108327424000000\) \([2]\) \(3057647616\) \(5.3448\)  
463680.fh5 463680fh5 \([0, 0, 0, -27254578188, -1725900003588112]\) \(11715873038622856702991202049/46415372499833400000000\) \(8870115816866722244198400000000\) \([2]\) \(1019215872\) \(4.7955\)  
463680.fh6 463680fh2 \([0, 0, 0, -2530630668, 1938013636592]\) \(9378698233516887309850369/5418996968417034240000\) \(1035586447599473252357898240000\) \([2, 2]\) \(509607936\) \(4.4489\)  
463680.fh7 463680fh1 \([0, 0, 0, -1775655948, 28723610732528]\) \(3239908336204082689644289/9880281924658790400\) \(1888151279521302630000230400\) \([2]\) \(254803968\) \(4.1023\) \(\Gamma_0(N)\)-optimal*
463680.fh8 463680fh4 \([0, 0, 0, 10113721332, 15497816721392]\) \(598672364899527954087397631/346996861747253448998400\) \(-66312132942560693929858459238400\) \([2]\) \(1019215872\) \(4.7955\)  
*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 463680.fh1.

Rank

sage: E.rank()
 

The elliptic curves in class 463680.fh have rank \(0\).

Complex multiplication

The elliptic curves in class 463680.fh do not have complex multiplication.

Modular form 463680.2.a.fh

sage: E.q_eigenform(10)
 
\(q - q^{5} + q^{7} - 2 q^{13} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 3 & 6 & 12 & 12 \\ 2 & 1 & 2 & 2 & 6 & 3 & 6 & 6 \\ 4 & 2 & 1 & 4 & 12 & 6 & 3 & 12 \\ 4 & 2 & 4 & 1 & 12 & 6 & 12 & 3 \\ 3 & 6 & 12 & 12 & 1 & 2 & 4 & 4 \\ 6 & 3 & 6 & 6 & 2 & 1 & 2 & 2 \\ 12 & 6 & 3 & 12 & 4 & 2 & 1 & 4 \\ 12 & 6 & 12 & 3 & 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.