# Properties

 Label 139650.ef Number of curves $6$ Conductor $139650$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("ef1")

sage: E.isogeny_class()

## Elliptic curves in class 139650.ef

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
139650.ef1 139650ha5 $$[1, 0, 1, -435505901, 3497436384698]$$ $$4969327007303723277361/1123462695162150$$ $$2065222853486434146093750$$ $$[2]$$ $$70778880$$ $$3.6579$$
139650.ef2 139650ha3 $$[1, 0, 1, -30337151, 41346947198]$$ $$1679731262160129361/570261564022500$$ $$1048292230401298476562500$$ $$[2, 2]$$ $$35389440$$ $$3.3113$$
139650.ef3 139650ha2 $$[1, 0, 1, -12476651, -16485351802]$$ $$116844823575501841/3760263939600$$ $$6912363941093756250000$$ $$[2, 2]$$ $$17694720$$ $$2.9647$$
139650.ef4 139650ha1 $$[1, 0, 1, -12378651, -16764259802]$$ $$114113060120923921/124104960$$ $$228137881860000000$$ $$[2]$$ $$8847360$$ $$2.6181$$ $$\Gamma_0(N)$$-optimal
139650.ef5 139650ha4 $$[1, 0, 1, 3815849, -56467146802]$$ $$3342636501165359/751262567039460$$ $$-1381020152337897336562500$$ $$[2]$$ $$35389440$$ $$3.3113$$
139650.ef6 139650ha6 $$[1, 0, 1, 89063599, 286596087698]$$ $$42502666283088696719/43898058864843750$$ $$-80696292615468786621093750$$ $$[2]$$ $$70778880$$ $$3.6579$$

## Rank

sage: E.rank()

The elliptic curves in class 139650.ef have rank $$0$$.

## Complex multiplication

The elliptic curves in class 139650.ef do not have complex multiplication.

## Modular form 139650.2.a.ef

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} + 4 q^{11} + q^{12} + 6 q^{13} + q^{16} + 2 q^{17} - q^{18} + q^{19} + O(q^{20})$$

## 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.