# Properties

 Label 27a Number of curves $4$ Conductor $27$ CM $$\Q(\sqrt{-3})$$ Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 27a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
27.a3 27a1 $$[0, 0, 1, 0, -7]$$ $$0$$ $$-19683$$ $$$$ $$1$$ $$-0.49716$$ $$\Gamma_0(N)$$-optimal $$-3$$
27.a1 27a2 $$[0, 0, 1, -270, -1708]$$ $$-12288000$$ $$-177147$$ $$[]$$ $$3$$ $$0.052148$$   $$-27$$
27.a4 27a3 $$[0, 0, 1, 0, 0]$$ $$0$$ $$-27$$ $$$$ $$3$$ $$-1.0465$$   $$-3$$
27.a2 27a4 $$[0, 0, 1, -30, 63]$$ $$-12288000$$ $$-243$$ $$$$ $$9$$ $$-0.49716$$   $$-27$$

## Rank

sage: E.rank()

The elliptic curves in class 27a have rank $$0$$.

## Complex multiplication

Each elliptic curve in class 27a has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-3})$$.

## Modular form27.2.a.a

sage: E.q_eigenform(10)

$$q - 2q^{4} - q^{7} + 5q^{13} + 4q^{16} - 7q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrrr} 1 & 3 & 3 & 9 \\ 3 & 1 & 9 & 27 \\ 3 & 9 & 1 & 3 \\ 9 & 27 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 