# Properties

 Label 2233b Number of curves $2$ Conductor $2233$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 2233b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2233.b1 2233b1 $$[0, 1, 1, -63, 271]$$ $$-28094464000/20657483$$ $$-20657483$$ $$$$ $$336$$ $$0.10307$$ $$\Gamma_0(N)$$-optimal
2233.b2 2233b2 $$[0, 1, 1, 517, -4340]$$ $$15252992000000/17621717267$$ $$-17621717267$$ $$[]$$ $$1008$$ $$0.65238$$

## Rank

sage: E.rank()

The elliptic curves in class 2233b have rank $$1$$.

## Complex multiplication

The elliptic curves in class 2233b do not have complex multiplication.

## Modular form2233.2.a.b

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{4} + q^{7} - 2q^{9} - q^{11} - 2q^{12} + 2q^{13} + 4q^{16} + 6q^{17} - 7q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 