# Properties

 Label 19074.x Number of curves 4 Conductor 19074 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("19074.x1")

sage: E.isogeny_class()

## Elliptic curves in class 19074.x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
19074.x1 19074p3 [1, 1, 1, -2908791, -1910699799]  512000
19074.x2 19074p4 [1, 1, 1, -2905901, -1914682219]  1024000
19074.x3 19074p1 [1, 1, 1, -13011, 410961]  102400 $$\Gamma_0(N)$$-optimal
19074.x4 19074p2 [1, 1, 1, 33229, 2722961]  204800

## Rank

sage: E.rank()

The elliptic curves in class 19074.x have rank $$0$$.

## Modular form 19074.2.a.x

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 