Properties

Label 450840g
Number of curves $6$
Conductor $450840$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("g1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 450840g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450840.g4 450840g1 \([0, -1, 0, -4790271, 4037012496]\) \(31476797652269056/49725\) \(19203849896400\) \([2]\) \(6488064\) \(2.2433\) \(\Gamma_0(N)\)-optimal
450840.g3 450840g2 \([0, -1, 0, -4791716, 4034456580]\) \(1969080716416336/2472575625\) \(15278582977575840000\) \([2, 2]\) \(12976128\) \(2.5898\)  
450840.g5 450840g3 \([0, -1, 0, -3514336, 6233083036]\) \(-194204905090564/566398828125\) \(-13999606574475600000000\) \([2]\) \(25952256\) \(2.9364\)  
450840.g2 450840g4 \([0, -1, 0, -6092216, 1672228380]\) \(1011710313226084/536724738225\) \(13266155932582784025600\) \([2, 2]\) \(25952256\) \(2.9364\)  
450840.g6 450840g5 \([0, -1, 0, 23212384, 13054135020]\) \(27980756504588158/17683545112935\) \(-874163794591910615070720\) \([2]\) \(51904512\) \(3.2830\)  
450840.g1 450840g6 \([0, -1, 0, -56204816, -160913091060]\) \(397210600760070242/3536192675535\) \(174807233951786341201920\) \([2]\) \(51904512\) \(3.2830\)  

Rank

sage: E.rank()
 

The elliptic curves in class 450840g have rank \(1\).

Complex multiplication

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

Modular form 450840.2.a.g

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} - 4q^{11} + q^{13} + q^{15} - 4q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.