# Properties

 Label 32487.f Number of curves $6$ Conductor $32487$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("f1")

E.isogeny_class()

## Elliptic curves in class 32487.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
32487.f1 32487l6 $$[1, 0, 0, -988527, 378154692]$$ $$908031902324522977/161726530797$$ $$19026964621736253$$ $$[2]$$ $$393216$$ $$2.1281$$
32487.f2 32487l4 $$[1, 0, 0, -68062, 4629995]$$ $$296380748763217/92608836489$$ $$10895337004094361$$ $$[2, 2]$$ $$196608$$ $$1.7815$$
32487.f3 32487l2 $$[1, 0, 0, -26657, -1622160]$$ $$17806161424897/668584449$$ $$78658291840401$$ $$[2, 2]$$ $$98304$$ $$1.4350$$
32487.f4 32487l1 $$[1, 0, 0, -26412, -1654353]$$ $$17319700013617/25857$$ $$3042050193$$ $$[2]$$ $$49152$$ $$1.0884$$ $$\Gamma_0(N)$$-optimal
32487.f5 32487l3 $$[1, 0, 0, 10828, -5812983]$$ $$1193377118543/124806800313$$ $$-14683395250024137$$ $$[2]$$ $$196608$$ $$1.7815$$
32487.f6 32487l5 $$[1, 0, 0, 189923, 31408838]$$ $$6439735268725823/7345472585373$$ $$-864187504196548077$$ $$[2]$$ $$393216$$ $$2.1281$$

## Rank

sage: E.rank()

The elliptic curves in class 32487.f have rank $$1$$.

## Complex multiplication

The elliptic curves in class 32487.f do not have complex multiplication.

## Modular form 32487.2.a.f

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} - q^{4} + 2 q^{5} - q^{6} + 3 q^{8} + q^{9} - 2 q^{10} + 4 q^{11} - q^{12} - q^{13} + 2 q^{15} - q^{16} - q^{17} - q^{18} + 4 q^{19} + O(q^{20})$$

## 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.