# Properties

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

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

sage: E.isogeny_class()

## Elliptic curves in class 19a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
19.a2 19a1 $$[0, 1, 1, -9, -15]$$ $$-89915392/6859$$ $$-6859$$ $$$$ $$1$$ $$-0.51587$$ $$\Gamma_0(N)$$-optimal
19.a1 19a2 $$[0, 1, 1, -769, -8470]$$ $$-50357871050752/19$$ $$-19$$ $$[]$$ $$3$$ $$0.033439$$
19.a3 19a3 $$[0, 1, 1, 1, 0]$$ $$32768/19$$ $$-19$$ $$$$ $$3$$ $$-1.0652$$

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 19a do not have complex multiplication.

## Modular form19.2.a.a

sage: E.q_eigenform(10)

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

## 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. 