# Properties

 Label 2205.b Number of curves 4 Conductor 2205 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("2205.b1")

sage: E.isogeny_class()

## Elliptic curves in class 2205.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2205.b1 2205k3 [1, -1, 1, -49622, -4241856]  6144
2205.b2 2205k2 [1, -1, 1, -3317, -55884] [2, 2] 3072
2205.b3 2205k1 [1, -1, 1, -1112, 13794]  1536 $$\Gamma_0(N)$$-optimal
2205.b4 2205k4 [1, -1, 1, 7708, -355764]  6144

## Rank

sage: E.rank()

The elliptic curves in class 2205.b have rank $$0$$.

## Modular form2205.2.a.b

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} + q^{5} + 3q^{8} - q^{10} + 6q^{13} - q^{16} + 2q^{17} + 8q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 