Properties

Label 450450o
Number of curves $4$
Conductor $450450$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("o1")
 
E.isogeny_class()
 

Elliptic curves in class 450450o

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450450.o4 450450o1 \([1, -1, 0, -543065292, -4870353806384]\) \(1555006827939811751684089/221961497899581440\) \(2528280187012419840000000\) \([2]\) \(139345920\) \(3.6982\) \(\Gamma_0(N)\)-optimal*
450450.o3 450450o2 \([1, -1, 0, -592457292, -3931560062384]\) \(2019051077229077416165369/582160888682835862400\) \(6631176372652927245150000000\) \([2]\) \(278691840\) \(4.0448\) \(\Gamma_0(N)\)-optimal*
450450.o2 450450o3 \([1, -1, 0, -1256768667, 10259897608741]\) \(19272683606216463573689449/7161126378530668544000\) \(81569705155450896384000000000\) \([2]\) \(418037760\) \(4.2475\) \(\Gamma_0(N)\)-optimal*
450450.o1 450450o4 \([1, -1, 0, -17771840667, 911669042440741]\) \(54497099771831721530744218729/16209843781074944000000\) \(184640251818806784000000000000\) \([2]\) \(836075520\) \(4.5941\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 450450o1.

Rank

sage: E.rank()
 

The elliptic curves in class 450450o have rank \(0\).

Complex multiplication

The elliptic curves in class 450450o do not have complex multiplication.

Modular form 450450.2.a.o

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{7} - q^{8} - q^{11} - q^{13} + q^{14} + q^{16} + 2 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.