# Properties

 Label 33.a Number of curves $4$ Conductor $33$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 33.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
33.a1 33a3 $$[1, 1, 0, -146, 621]$$ $$347873904937/395307$$ $$395307$$ $$$$ $$6$$ $$-0.012632$$
33.a2 33a1 $$[1, 1, 0, -11, 0]$$ $$169112377/88209$$ $$88209$$ $$[2, 2]$$ $$3$$ $$-0.35921$$ $$\Gamma_0(N)$$-optimal
33.a3 33a2 $$[1, 1, 0, -6, -9]$$ $$30664297/297$$ $$297$$ $$$$ $$6$$ $$-0.70578$$
33.a4 33a4 $$[1, 1, 0, 44, 55]$$ $$9090072503/5845851$$ $$-5845851$$ $$$$ $$6$$ $$-0.012632$$

## Rank

sage: E.rank()

The elliptic curves in class 33.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 33.a do not have complex multiplication.

## Modular form33.2.a.a

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} - 2q^{5} - q^{6} + 4q^{7} - 3q^{8} + q^{9} - 2q^{10} + q^{11} + q^{12} - 2q^{13} + 4q^{14} + 2q^{15} - q^{16} - 2q^{17} + q^{18} + O(q^{20})$$ ## 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. 