# Properties

 Label 140679.z Number of curves $4$ Conductor $140679$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("z1")

sage: E.isogeny_class()

## Elliptic curves in class 140679.z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
140679.z1 140679y3 $$[1, -1, 0, -23529858, 43932670109]$$ $$16798320881842096017/2132227789307$$ $$182872906577266668147$$ $$[2]$$ $$8257536$$ $$2.9098$$
140679.z2 140679y4 $$[1, -1, 0, -9334068, -10525526425]$$ $$1048626554636928177/48569076788309$$ $$4165581316684401079389$$ $$[2]$$ $$8257536$$ $$2.9098$$
140679.z3 140679y2 $$[1, -1, 0, -1596723, 562088960]$$ $$5249244962308257/1448621666569$$ $$124242661138178508849$$ $$[2, 2]$$ $$4128768$$ $$2.5632$$
140679.z4 140679y1 $$[1, -1, 0, 257682, 57319919]$$ $$22062729659823/29354283343$$ $$-2517603017064022503$$ $$[2]$$ $$2064384$$ $$2.2166$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 140679.z have rank $$0$$.

## Complex multiplication

The elliptic curves in class 140679.z do not have complex multiplication.

## Modular form 140679.2.a.z

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - 2q^{5} - 3q^{8} - 2q^{10} + q^{11} - 6q^{13} - q^{16} - 2q^{17} + 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.