# Properties

 Label 19074i Number of curves 2 Conductor 19074 CM no Rank 2 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("19074.f1")

sage: E.isogeny_class()

## Elliptic curves in class 19074i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
19074.f2 19074i1 [1, 0, 1, -117, 460]  6144 $$\Gamma_0(N)$$-optimal
19074.f1 19074i2 [1, 0, 1, -287, -1240]  12288

## Rank

sage: E.rank()

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

## Modular form 19074.2.a.f

sage: E.q_eigenform(10)

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

## 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 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. 