# Properties

 Label 108.a Number of curves $2$ Conductor $108$ CM $$\Q(\sqrt{-3})$$ Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 108.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
108.a1 108a2 $$[0, 0, 0, 0, -108]$$ $$0$$ $$-5038848$$ $$[]$$ $$18$$ $$-0.035060$$   $$-3$$
108.a2 108a1 $$[0, 0, 0, 0, 4]$$ $$0$$ $$-6912$$ $$$$ $$6$$ $$-0.58437$$ $$\Gamma_0(N)$$-optimal $$-3$$

## Rank

sage: E.rank()

The elliptic curves in class 108.a have rank $$0$$.

## Complex multiplication

Each elliptic curve in class 108.a has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-3})$$.

## Modular form108.2.a.a

sage: E.q_eigenform(10)

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

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 