# Properties

 Label 19a Number of curves $3$ Conductor $19$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("19.a1")

sage: E.isogeny_class()

## Elliptic curves in class 19a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
19.a2 19a1 [0, 1, 1, -9, -15]  1 $$\Gamma_0(N)$$-optimal
19.a1 19a2 [0, 1, 1, -769, -8470] [] 3
19.a3 19a3 [0, 1, 1, 1, 0]  3

## Rank

sage: E.rank()

The elliptic curves in class 19a have rank $$0$$.

## Modular form19.2.a.a

sage: E.q_eigenform(10)

$$q - 2q^{3} - 2q^{4} + 3q^{5} - q^{7} + q^{9} + 3q^{11} + 4q^{12} - 4q^{13} - 6q^{15} + 4q^{16} - 3q^{17} + q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrr} 1 & 3 & 3 \\ 3 & 1 & 9 \\ 3 & 9 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 