# Properties

 Label 19220b Number of curves 4 Conductor 19220 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("19220.c1")

sage: E.isogeny_class()

## Elliptic curves in class 19220b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
19220.c3 19220b1 [0, -1, 0, -1281, -11710]  15120 $$\Gamma_0(N)$$-optimal
19220.c4 19220b2 [0, -1, 0, 3524, -82824]  30240
19220.c1 19220b3 [0, -1, 0, -39721, 3059646]  45360
19220.c2 19220b4 [0, -1, 0, -34916, 3822680]  90720

## Rank

sage: E.rank()

The elliptic curves in class 19220b have rank $$1$$.

## Modular form 19220.2.a.c

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 