# Properties

 Label 8112.be Number of curves $6$ Conductor $8112$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("be1")

sage: E.isogeny_class()

## Elliptic curves in class 8112.be

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8112.be1 8112k5 $$[0, 1, 0, -64952, 6349812]$$ $$3065617154/9$$ $$88967743488$$ $$[2]$$ $$18432$$ $$1.3303$$
8112.be2 8112k3 $$[0, 1, 0, -10872, -439932]$$ $$28756228/3$$ $$14827957248$$ $$[2]$$ $$9216$$ $$0.98370$$
8112.be3 8112k4 $$[0, 1, 0, -4112, 95460]$$ $$1556068/81$$ $$400354845696$$ $$[2, 2]$$ $$9216$$ $$0.98370$$
8112.be4 8112k2 $$[0, 1, 0, -732, -5940]$$ $$35152/9$$ $$11120967936$$ $$[2, 2]$$ $$4608$$ $$0.63712$$
8112.be5 8112k1 $$[0, 1, 0, 113, -532]$$ $$2048/3$$ $$-231686832$$ $$[2]$$ $$2304$$ $$0.29055$$ $$\Gamma_0(N)$$-optimal
8112.be6 8112k6 $$[0, 1, 0, 2648, 384788]$$ $$207646/6561$$ $$-64857485002752$$ $$[2]$$ $$18432$$ $$1.3303$$

## Rank

sage: E.rank()

The elliptic curves in class 8112.be have rank $$0$$.

## Complex multiplication

The elliptic curves in class 8112.be do not have complex multiplication.

## Modular form8112.2.a.be

sage: E.q_eigenform(10)

$$q + q^{3} + 2q^{5} + q^{9} + 4q^{11} + 2q^{15} + 2q^{17} - 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

$$\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.