# Properties

 Label 19074.p Number of curves $2$ Conductor $19074$ CM no Rank $1$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("p1")

sage: E.isogeny_class()

## Elliptic curves in class 19074.p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
19074.p1 19074x2 $$[1, 1, 1, -52604, 4621691]$$ $$666940371553/37026$$ $$893717629794$$ $$$$ $$55296$$ $$1.3586$$
19074.p2 19074x1 $$[1, 1, 1, -3474, 62427]$$ $$192100033/38148$$ $$920799982212$$ $$$$ $$27648$$ $$1.0120$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 19074.p do not have complex multiplication.

## Modular form 19074.2.a.p

sage: E.q_eigenform(10)

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