# Properties

 Label 19074b Number of curves 4 Conductor 19074 CM no Rank 2 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("19074.a1")

sage: E.isogeny_class()

## Elliptic curves in class 19074b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
19074.a3 19074b1 [1, 1, 0, -1595, 22473] [2] 18432 $$\Gamma_0(N)$$-optimal
19074.a4 19074b2 [1, 1, 0, 1295, 98191] [2] 36864
19074.a1 19074b3 [1, 1, 0, -23270, -1370796] [2] 55296
19074.a2 19074b4 [1, 1, 0, -11710, -2718692] [2] 110592

## Rank

sage: E.rank()

The elliptic curves in class 19074b have rank $$2$$.

## Modular form 19074.2.a.a

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.