# Properties

 Label 19074y Number of curves 4 Conductor 19074 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("19074.m1")

sage: E.isogeny_class()

## Elliptic curves in class 19074y

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
19074.m4 19074y1 [1, 1, 1, -1689789, -661874733]  774144 $$\Gamma_0(N)$$-optimal
19074.m2 19074y2 [1, 1, 1, -25364669, -49176438829] [2, 2] 1548288
19074.m1 19074y3 [1, 1, 1, -405827389, -3146903905069]  3096576
19074.m3 19074y4 [1, 1, 1, -23700029, -55907577133]  3096576

## Rank

sage: E.rank()

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

## Modular form 19074.2.a.m

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 