# Properties

 Label 19074r Number of curves $2$ Conductor $19074$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("r1")

sage: E.isogeny_class()

## Elliptic curves in class 19074r

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
19074.l1 19074r1 $$[1, 1, 1, -56650, -5210089]$$ $$832972004929/610368$$ $$14732799715392$$ $$[2]$$ $$110592$$ $$1.4610$$ $$\Gamma_0(N)$$-optimal
19074.l2 19074r2 $$[1, 1, 1, -45090, -7383369]$$ $$-420021471169/727634952$$ $$-17563338860711688$$ $$[2]$$ $$221184$$ $$1.8075$$

## Rank

sage: E.rank()

The elliptic curves in class 19074r have rank $$0$$.

## Complex multiplication

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

## Modular form 19074.2.a.r

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.