# Properties

 Label 4002.h Number of curves 2 Conductor 4002 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("4002.h1")
sage: E.isogeny_class()

## Elliptic curves in class 4002.h

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
4002.h1 4002k2 [1, 1, 1, -92909, -10938229] 2 16128
4002.h2 4002k1 [1, 1, 1, -5429, -195685] 2 8064 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 4002.h have rank $$1$$.

## Modular form4002.2.a.h

sage: E.q_eigenform(10)
$$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - 2q^{7} + q^{8} + q^{9} - 2q^{10} + 2q^{11} - q^{12} - 2q^{13} - 2q^{14} + 2q^{15} + q^{16} + q^{18} + 4q^{19} + O(q^{20})$$

## 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 LMFDB 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 LMFDB labels.