# Properties

 Label 1045.a Number of curves $2$ Conductor $1045$ CM no Rank $1$ Graph # Related objects

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

E.isogeny_class()

## Elliptic curves in class 1045.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1045.a1 1045a2 $$[1, 0, 0, -31, 60]$$ $$3301293169/218405$$ $$218405$$ $$$$ $$128$$ $$-0.22604$$
1045.a2 1045a1 $$[1, 0, 0, -6, -5]$$ $$24137569/5225$$ $$5225$$ $$$$ $$64$$ $$-0.57262$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 1045.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 1045.a do not have complex multiplication.

## Modular form1045.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - 2 q^{3} - q^{4} - q^{5} + 2 q^{6} - 2 q^{7} + 3 q^{8} + q^{9} + q^{10} - q^{11} + 2 q^{12} + 6 q^{13} + 2 q^{14} + 2 q^{15} - q^{16} - q^{18} - q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 