Properties

Label 450840q
Number of curves $4$
Conductor $450840$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("q1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 450840q

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450840.q4 450840q1 \([0, -1, 0, -3037775, -414540048]\) \(8027441608013824/4452347908125\) \(1719501677509965570000\) \([2]\) \(28311552\) \(2.7651\) \(\Gamma_0(N)\)-optimal
450840.q2 450840q2 \([0, -1, 0, -36863780, -86007863100]\) \(896581610757188944/1545359765625\) \(9549114360984900000000\) \([2, 2]\) \(56623104\) \(3.1117\)  
450840.q3 450840q3 \([0, -1, 0, -25367360, -140643449508]\) \(-73039208963041156/303497314453125\) \(-7501503865781250000000000\) \([4]\) \(113246208\) \(3.4583\)  
450840.q1 450840q4 \([0, -1, 0, -589576280, -5509886168100]\) \(916959671620739147236/2731145625\) \(67505373155825280000\) \([2]\) \(113246208\) \(3.4583\)  

Rank

sage: E.rank()
 

The elliptic curves in class 450840q have rank \(0\).

Complex multiplication

The elliptic curves in class 450840q do not have complex multiplication.

Modular form 450840.2.a.q

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{5} - 4q^{7} + q^{9} + q^{13} - q^{15} + 4q^{19} + O(q^{20})\)  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 Cremona 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 Cremona labels.