# Properties

 Label 10710bm Number of curves $2$ Conductor $10710$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bm1")

sage: E.isogeny_class()

## Elliptic curves in class 10710bm

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10710.bj2 10710bm1 $$[1, -1, 1, 733, -9309]$$ $$59822347031/83966400$$ $$-61211505600$$ $$[2]$$ $$9216$$ $$0.75542$$ $$\Gamma_0(N)$$-optimal
10710.bj1 10710bm2 $$[1, -1, 1, -4667, -89229]$$ $$15417797707369/4080067320$$ $$2974369076280$$ $$[2]$$ $$18432$$ $$1.1020$$

## Rank

sage: E.rank()

The elliptic curves in class 10710bm have rank $$1$$.

## Complex multiplication

The elliptic curves in class 10710bm do not have complex multiplication.

## Modular form 10710.2.a.bm

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} + q^{10} - 2q^{11} - 2q^{13} + q^{14} + q^{16} - q^{17} - 4q^{19} + O(q^{20})$$

## 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 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.