Properties

Label 405600fh
Number of curves $4$
Conductor $405600$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

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

Elliptic curves in class 405600fh

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405600.fh3 405600fh1 \([0, 1, 0, -6766955758, 214241224047488]\) \(7099759044484031233216/577161945398025\) \(2785850472504695652225000000\) \([2, 2]\) \(433520640\) \(4.3113\) \(\Gamma_0(N)\)-optimal*
405600.fh1 405600fh2 \([0, 1, 0, -108269179633, 13712108457168863]\) \(454357982636417669333824/3003024375\) \(927681605150040000000000\) \([2]\) \(867041280\) \(4.6579\) \(\Gamma_0(N)\)-optimal*
405600.fh4 405600fh3 \([0, 1, 0, -6304952008, 244750103682488]\) \(-717825640026599866952/254764560814329735\) \(-9837599000157232750924920000000\) \([2]\) \(867041280\) \(4.6579\)  
405600.fh2 405600fh4 \([0, 1, 0, -7231072008, 183170497574988]\) \(1082883335268084577352/251301565117746585\) \(9703877249795402209578120000000\) \([2]\) \(867041280\) \(4.6579\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 405600fh1.

Rank

sage: E.rank()
 

The elliptic curves in class 405600fh have rank \(0\).

Complex multiplication

The elliptic curves in class 405600fh do not have complex multiplication.

Modular form 405600.2.a.fh

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{9} + 4 q^{11} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.