# Properties

 Label 47040.gp Number of curves $8$ Conductor $47040$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("gp1")

sage: E.isogeny_class()

## Elliptic curves in class 47040.gp

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
47040.gp1 47040dc8 $$[0, 1, 0, -1101465185, 14069962595583]$$ $$4791901410190533590281/41160000$$ $$1269414714408960000$$ $$[2]$$ $$10616832$$ $$3.5149$$
47040.gp2 47040dc6 $$[0, 1, 0, -68843105, 219815685375]$$ $$1169975873419524361/108425318400$$ $$3343943017284658790400$$ $$[2, 2]$$ $$5308416$$ $$3.1684$$
47040.gp3 47040dc7 $$[0, 1, 0, -63825505, 253221862655]$$ $$-932348627918877961/358766164249920$$ $$-11064699901139704369643520$$ $$[2]$$ $$10616832$$ $$3.5149$$
47040.gp4 47040dc5 $$[0, 1, 0, -13665185, 19096955583]$$ $$9150443179640281/184570312500$$ $$5692329216000000000000$$ $$[2]$$ $$3538944$$ $$2.9656$$
47040.gp5 47040dc3 $$[0, 1, 0, -4617825, 2901224703]$$ $$353108405631241/86318776320$$ $$2662155607152179281920$$ $$[2]$$ $$2654208$$ $$2.8218$$
47040.gp6 47040dc2 $$[0, 1, 0, -1811105, -493097025]$$ $$21302308926361/8930250000$$ $$275417656786944000000$$ $$[2, 2]$$ $$1769472$$ $$2.6191$$
47040.gp7 47040dc1 $$[0, 1, 0, -1560225, -750349377]$$ $$13619385906841/6048000$$ $$186526243749888000$$ $$[2]$$ $$884736$$ $$2.2725$$ $$\Gamma_0(N)$$-optimal
47040.gp8 47040dc4 $$[0, 1, 0, 6028895, -3614985025]$$ $$785793873833639/637994920500$$ $$-19676388236172853248000$$ $$[2]$$ $$3538944$$ $$2.9656$$

## Rank

sage: E.rank()

The elliptic curves in class 47040.gp have rank $$0$$.

## Complex multiplication

The elliptic curves in class 47040.gp do not have complex multiplication.

## Modular form 47040.2.a.gp

sage: E.q_eigenform(10)

$$q + q^{3} + q^{5} + q^{9} + 2q^{13} + q^{15} + 6q^{17} + 8q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.