# Properties

 Label 99b Number of curves 4 Conductor 99 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("99.b1")

sage: E.isogeny_class()

## Elliptic curves in class 99b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
99.b3 99b1 [1, -1, 1, -59, 186]  12 $$\Gamma_0(N)$$-optimal
99.b2 99b2 [1, -1, 1, -104, -102] [2, 2] 24
99.b1 99b3 [1, -1, 1, -1319, -18084]  48
99.b4 99b4 [1, -1, 1, 391, -1092]  48

## Rank

sage: E.rank()

The elliptic curves in class 99b have rank $$0$$.

## Modular form99.2.a.b

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 