# Properties

 Label 19074t Number of curves 2 Conductor 19074 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("19074.w1")

sage: E.isogeny_class()

## Elliptic curves in class 19074t

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
19074.w1 19074t1 [1, 1, 1, -261262, 50884811]  322560 $$\Gamma_0(N)$$-optimal
19074.w2 19074t2 [1, 1, 1, -76302, 121761483]  645120

## Rank

sage: E.rank()

The elliptic curves in class 19074t have rank $$1$$.

## Modular form 19074.2.a.w

sage: E.q_eigenform(10)

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

## 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. 