# Properties

 Label 273999h Number of curves 2 Conductor 273999 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("273999.h1")

sage: E.isogeny_class()

## Elliptic curves in class 273999h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
273999.h2 273999h1 [1, 0, 1, -8311, 4239749]  1140480 $$\Gamma_0(N)$$-optimal
273999.h1 273999h2 [1, 0, 1, -446926, 114068945]  2280960

## Rank

sage: E.rank()

The elliptic curves in class 273999h have rank $$1$$.

## Modular form 273999.2.a.h

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} - q^{4} + q^{6} - 2q^{7} - 3q^{8} + q^{9} + q^{11} - q^{12} - 2q^{13} - 2q^{14} - q^{16} + q^{18} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 