# Properties

 Label 112710cz Number of curves $2$ Conductor $112710$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("cz1")

sage: E.isogeny_class()

## Elliptic curves in class 112710cz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
112710.cy2 112710cz1 $$[1, 0, 0, 1150, -41568]$$ $$6967871/35100$$ $$-847228671900$$ $$[2]$$ $$199680$$ $$0.97099$$ $$\Gamma_0(N)$$-optimal
112710.cy1 112710cz2 $$[1, 0, 0, -13300, -529978]$$ $$10779215329/1232010$$ $$29737726383690$$ $$[2]$$ $$399360$$ $$1.3176$$

## Rank

sage: E.rank()

The elliptic curves in class 112710cz have rank $$0$$.

## Complex multiplication

The elliptic curves in class 112710cz do not have complex multiplication.

## Modular form 112710.2.a.cz

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - 2q^{7} + q^{8} + q^{9} + q^{10} - 4q^{11} + q^{12} - q^{13} - 2q^{14} + q^{15} + q^{16} + q^{18} - 6q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.