# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("10005.l1")
sage: E.isogeny_class()

## Elliptic curves in class 10005g

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
10005.l2 10005g1 [1, 1, 0, -6747, -248544] 2 33024 $$\Gamma_0(N)$$-optimal
10005.l1 10005g2 [1, 1, 0, -112452, -14561001] 2 66048

## Rank

sage: E.rank()

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

## Modular form None

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

## 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.