# Properties

 Label 465690.bq Number of curves $4$ Conductor $465690$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bq1")

sage: E.isogeny_class()

## Elliptic curves in class 465690.bq

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
465690.bq1 465690bq4 [1, 1, 1, -302706, -63792231] [2] 5308416
465690.bq2 465690bq2 [1, 1, 1, -31956, 537969] [2, 2] 2654208
465690.bq3 465690bq1 [1, 1, 1, -24736, 1485233] [2] 1327104 $$\Gamma_0(N)$$-optimal*
465690.bq4 465690bq3 [1, 1, 1, 123274, 4387673] [2] 5308416
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 465690.bq1.

## Rank

sage: E.rank()

The elliptic curves in class 465690.bq have rank $$1$$.

## Complex multiplication

The elliptic curves in class 465690.bq do not have complex multiplication.

## Modular form 465690.2.a.bq

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + 4q^{7} + q^{8} + q^{9} - q^{10} - q^{12} + 2q^{13} + 4q^{14} + q^{15} + q^{16} + 2q^{17} + q^{18} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.