# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("19074.d1")

sage: E.isogeny_class()

## Elliptic curves in class 19074a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
19074.d2 19074a1 [1, 1, 0, -157944, 23574528]  193536 $$\Gamma_0(N)$$-optimal
19074.d1 19074a2 [1, 1, 0, -354464, -47054760]  387072

## Rank

sage: E.rank()

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

## Modular form 19074.2.a.d

sage: E.q_eigenform(10)

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

## 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. 