# Properties

 Label 44.a Number of curves $2$ Conductor $44$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 44.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
44.a1 44a2 $$[0, 1, 0, -77, -289]$$ $$-199794688/1331$$ $$-340736$$ $$[]$$ $$6$$ $$-0.10299$$
44.a2 44a1 $$[0, 1, 0, 3, -1]$$ $$8192/11$$ $$-2816$$ $$$$ $$2$$ $$-0.65229$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 44.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 44.a do not have complex multiplication.

## Modular form44.2.a.a

sage: E.q_eigenform(10)

$$q + q^{3} - 3q^{5} + 2q^{7} - 2q^{9} - q^{11} - 4q^{13} - 3q^{15} + 6q^{17} + 8q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 