# Properties

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

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 450840.g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450840.g1 450840g6 $$[0, -1, 0, -56204816, -160913091060]$$ $$397210600760070242/3536192675535$$ $$174807233951786341201920$$ $$[2]$$ $$51904512$$ $$3.2830$$
450840.g2 450840g4 $$[0, -1, 0, -6092216, 1672228380]$$ $$1011710313226084/536724738225$$ $$13266155932582784025600$$ $$[2, 2]$$ $$25952256$$ $$2.9364$$
450840.g3 450840g2 $$[0, -1, 0, -4791716, 4034456580]$$ $$1969080716416336/2472575625$$ $$15278582977575840000$$ $$[2, 2]$$ $$12976128$$ $$2.5898$$
450840.g4 450840g1 $$[0, -1, 0, -4790271, 4037012496]$$ $$31476797652269056/49725$$ $$19203849896400$$ $$[2]$$ $$6488064$$ $$2.2433$$ $$\Gamma_0(N)$$-optimal*
450840.g5 450840g3 $$[0, -1, 0, -3514336, 6233083036]$$ $$-194204905090564/566398828125$$ $$-13999606574475600000000$$ $$[2]$$ $$25952256$$ $$2.9364$$
450840.g6 450840g5 $$[0, -1, 0, 23212384, 13054135020]$$ $$27980756504588158/17683545112935$$ $$-874163794591910615070720$$ $$[2]$$ $$51904512$$ $$3.2830$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 450840.g1.

## Rank

sage: E.rank()

The elliptic curves in class 450840.g have rank $$1$$.

## Complex multiplication

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

## Modular form 450840.2.a.g

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} + q^{9} - 4q^{11} + q^{13} + q^{15} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.