# Properties

 Label 27690bc Number of curves 2 Conductor 27690 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("27690.bg1")
sage: E.isogeny_class()

## Elliptic curves in class 27690bc

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
27690.bg2 27690bc1 [1, 0, 0, -319040, 74649600] 7 526848 $$\Gamma_0(N)$$-optimal
27690.bg1 27690bc2 [1, 0, 0, -8462840, -15758963160] 1 3687936

## Rank

sage: E.rank()

The elliptic curves in class 27690bc have rank $$1$$.

## Modular form 27690.2.a.bg

sage: E.q_eigenform(10)
$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + q^{10} - 2q^{11} + q^{12} - q^{13} + q^{14} + q^{15} + q^{16} + 4q^{17} + q^{18} - 8q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 