# Properties

 Label 192027p Number of curves 2 Conductor 192027 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("192027.l1")

sage: E.isogeny_class()

## Elliptic curves in class 192027p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
192027.l2 192027p1 [1, 1, 0, -1473540, -10012407093]  10137600 $$\Gamma_0(N)$$-optimal
192027.l1 192027p2 [1, 1, 0, -79244475, -269440692066]  20275200

## Rank

sage: E.rank()

The elliptic curves in class 192027p have rank $$1$$.

## Modular form 192027.2.a.l

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} - q^{6} - 2q^{7} - 3q^{8} + q^{9} + q^{12} - 2q^{13} - 2q^{14} - q^{16} + q^{18} + 2q^{19} + O(q^{20})$$

## 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. 