# Properties

 Label 10005i Number of curves 2 Conductor 10005 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("10005.h1")

sage: E.isogeny_class()

## Elliptic curves in class 10005i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
10005.h2 10005i1 [0, 1, 1, -124021, 14938111] [3] 64800 $$\Gamma_0(N)$$-optimal
10005.h1 10005i2 [0, 1, 1, -2379061, -1411043480] [] 194400

## Rank

sage: E.rank()

The elliptic curves in class 10005i have rank $$1$$.

## Modular form 10005.2.a.h

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{4} - q^{5} - q^{7} + q^{9} - 2q^{12} - 4q^{13} - q^{15} + 4q^{16} - 3q^{17} - q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.