# Properties

 Label 465690bw Number of curves $2$ Conductor $465690$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("bw1")

sage: E.isogeny_class()

## Elliptic curves in class 465690bw

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
465690.bw2 465690bw1 $$[1, 1, 1, -13545, -412905]$$ $$5841725401/1857600$$ $$87392428545600$$ $$$$ $$1886976$$ $$1.3790$$ $$\Gamma_0(N)$$-optimal*
465690.bw1 465690bw2 $$[1, 1, 1, -85745, 9319655]$$ $$1481933914201/53916840$$ $$2536565238536040$$ $$$$ $$3773952$$ $$1.7255$$ $$\Gamma_0(N)$$-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 465690bw1.

## Rank

sage: E.rank()

The elliptic curves in class 465690bw have rank $$0$$.

## Complex multiplication

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

## Modular form 465690.2.a.bw

sage: E.q_eigenform(10)

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