# Properties

 Label 52800hf Number of curves 4 Conductor 52800 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("52800.hs1")
sage: E.isogeny_class()

## Elliptic curves in class 52800hf

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
52800.hs3 52800hf1 [0, 1, 0, -10433, 403263] 2 98304 $$\Gamma_0(N)$$-optimal
52800.hs2 52800hf2 [0, 1, 0, -18433, -308737] 4 196608
52800.hs4 52800hf3 [0, 1, 0, 69567, -2332737] 2 393216
52800.hs1 52800hf4 [0, 1, 0, -234433, -43724737] 2 393216

## Rank

sage: E.rank()

The elliptic curves in class 52800hf have rank $$0$$.

## Modular form 52800.2.a.hs

sage: E.q_eigenform(10)
$$q + q^{3} + 4q^{7} + q^{9} + q^{11} - 2q^{13} + 2q^{17} + O(q^{20})$$

## 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. 