Properties

Label 487872.lz
Number of curves $4$
Conductor $487872$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
sage: E = EllipticCurve("lz1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 487872.lz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
487872.lz1 487872lz3 \([0, 0, 0, -2788852044, -40195029294992]\) \(14171198121996897746/4077720290568771\) \(690257283227443598222439088128\) \([2]\) \(707788800\) \(4.4320\) \(\Gamma_0(N)\)-optimal*
487872.lz2 487872lz2 \([0, 0, 0, -2556938604, -49759417856720]\) \(21843440425782779332/3100814593569\) \(262445644211953132592234496\) \([2, 2]\) \(353894400\) \(4.0854\) \(\Gamma_0(N)\)-optimal*
487872.lz3 487872lz1 \([0, 0, 0, -2556851484, -49762978590512]\) \(87364831012240243408/1760913\) \(37259882261638299648\) \([2]\) \(176947200\) \(3.7388\) \(\Gamma_0(N)\)-optimal*
487872.lz4 487872lz4 \([0, 0, 0, -2326419084, -59095919455760]\) \(-8226100326647904626/4152140742401883\) \(-702854826766095092257341702144\) \([2]\) \(707788800\) \(4.4320\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 487872.lz1.

Rank

sage: E.rank()
 

The elliptic curves in class 487872.lz have rank \(1\).

Complex multiplication

The elliptic curves in class 487872.lz do not have complex multiplication.

Modular form 487872.2.a.lz

sage: E.q_eigenform(10)
 
\(q + 2q^{5} - q^{7} - 6q^{13} - 2q^{17} - 8q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.