# Properties

 Label 660.d Number of curves $4$ Conductor $660$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 660.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
660.d1 660d4 $$[0, 1, 0, -249996, -48194796]$$ $$6749703004355978704/5671875$$ $$1452000000$$ $$$$ $$1728$$ $$1.4936$$
660.d2 660d3 $$[0, 1, 0, -15621, -757296]$$ $$-26348629355659264/24169921875$$ $$-386718750000$$ $$$$ $$864$$ $$1.1470$$
660.d3 660d2 $$[0, 1, 0, -3156, -63900]$$ $$13584145739344/1195803675$$ $$306125740800$$ $$$$ $$576$$ $$0.94425$$
660.d4 660d1 $$[0, 1, 0, 219, -4500]$$ $$72268906496/606436875$$ $$-9702990000$$ $$$$ $$288$$ $$0.59767$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 660.d have rank $$0$$.

## Complex multiplication

The elliptic curves in class 660.d do not have complex multiplication.

## Modular form660.2.a.d

sage: E.q_eigenform(10)

$$q + q^{3} - q^{5} + 2q^{7} + q^{9} + q^{11} + 2q^{13} - q^{15} + 2q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 