# Properties

 Label 2205k Number of curves $4$ Conductor $2205$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("k1")

sage: E.isogeny_class()

## Elliptic curves in class 2205k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2205.b3 2205k1 $$[1, -1, 1, -1112, 13794]$$ $$1771561/105$$ $$9005442705$$ $$$$ $$1536$$ $$0.66315$$ $$\Gamma_0(N)$$-optimal
2205.b2 2205k2 $$[1, -1, 1, -3317, -55884]$$ $$47045881/11025$$ $$945571484025$$ $$[2, 2]$$ $$3072$$ $$1.0097$$
2205.b1 2205k3 $$[1, -1, 1, -49622, -4241856]$$ $$157551496201/13125$$ $$1125680338125$$ $$$$ $$6144$$ $$1.3563$$
2205.b4 2205k4 $$[1, -1, 1, 7708, -355764]$$ $$590589719/972405$$ $$-83399404891005$$ $$$$ $$6144$$ $$1.3563$$

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 2205k do not have complex multiplication.

## Modular form2205.2.a.k

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 Cremona 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 Cremona labels. 