# Properties

 Label 10005h Number of curves 2 Conductor 10005 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("10005.b1")

sage: E.isogeny_class()

## Elliptic curves in class 10005h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
10005.b2 10005h1 [1, 1, 1, -75, 360]  2944 $$\Gamma_0(N)$$-optimal
10005.b1 10005h2 [1, 1, 1, -1380, 19152]  5888

## Rank

sage: E.rank()

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

## Modular form 10005.2.a.b

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} - 4q^{7} + 3q^{8} + q^{9} - q^{10} + 2q^{11} + q^{12} + 2q^{13} + 4q^{14} - q^{15} - q^{16} + 4q^{17} - q^{18} - 6q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 