# Properties

 Label 20449g Number of curves 3 Conductor 20449 CM no Rank 2 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("20449.a1")

sage: E.isogeny_class()

## Elliptic curves in class 20449g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
20449.a3 20449g1 [0, -1, 1, -6816, -512172] [] 51840 $$\Gamma_0(N)$$-optimal
20449.a2 20449g2 [0, -1, 1, -211306, 67787488] [] 259200
20449.a1 20449g3 [0, -1, 1, -159917996, 778437705528] [] 1296000

## Rank

sage: E.rank()

The elliptic curves in class 20449g have rank $$2$$.

## Modular form 20449.2.a.a

sage: E.q_eigenform(10)

$$q - 2q^{2} - q^{3} + 2q^{4} - q^{5} + 2q^{6} - 2q^{7} - 2q^{9} + 2q^{10} - 2q^{12} + 4q^{14} + q^{15} - 4q^{16} + 2q^{17} + 4q^{18} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 