# Properties

 Label 99.a Number of curves $2$ Conductor $99$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 99.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
99.a1 99a2 $$[1, -1, 1, -17, 30]$$ $$19034163/121$$ $$3267$$ $$$$ $$8$$ $$-0.48869$$
99.a2 99a1 $$[1, -1, 1, -2, 0]$$ $$19683/11$$ $$297$$ $$$$ $$4$$ $$-0.83526$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 99.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 99.a do not have complex multiplication.

## Modular form99.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} - 4 q^{5} - 2 q^{7} + 3 q^{8} + 4 q^{10} - q^{11} - 2 q^{13} + 2 q^{14} - q^{16} + 2 q^{17} - 6 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 