# Properties

 Label 199920p Number of curves 4 Conductor 199920 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("199920.hc1")

sage: E.isogeny_class()

## Elliptic curves in class 199920p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
199920.hc4 199920p1 [0, 1, 0, 10960, 746388]  786432 $$\Gamma_0(N)$$-optimal
199920.hc3 199920p2 [0, 1, 0, -87040, 8076788] [2, 2] 1572864
199920.hc1 199920p3 [0, 1, 0, -1321840, 584481428]  3145728
199920.hc2 199920p4 [0, 1, 0, -420240, -97614252]  3145728

## Rank

sage: E.rank()

The elliptic curves in class 199920p have rank $$0$$.

## Modular form 199920.2.a.hc

sage: E.q_eigenform(10)

$$q + q^{3} + q^{5} + q^{9} + 6q^{13} + q^{15} + q^{17} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 