# Properties

 Label 1045b Number of curves $4$ Conductor $1045$ CM no Rank $1$ Graph

# Related objects

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

E.isogeny_class()

## Elliptic curves in class 1045b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1045.b3 1045b1 $$[1, -1, 0, -982409, -374543312]$$ $$104857852278310619039721/47155625$$ $$47155625$$ $$[2]$$ $$4416$$ $$1.7158$$ $$\Gamma_0(N)$$-optimal
1045.b2 1045b2 $$[1, -1, 0, -982414, -374539305]$$ $$104859453317683374662841/2223652969140625$$ $$2223652969140625$$ $$[2, 2]$$ $$8832$$ $$2.0624$$
1045.b1 1045b3 $$[1, -1, 0, -1016789, -346894930]$$ $$116256292809537371612841/15216540068579856875$$ $$15216540068579856875$$ $$[4]$$ $$17664$$ $$2.4090$$
1045.b4 1045b4 $$[1, -1, 0, -948119, -401927292]$$ $$-94256762600623910012361/15323275604248046875$$ $$-15323275604248046875$$ $$[2]$$ $$17664$$ $$2.4090$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form1045.2.a.b

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} + q^{5} - 3 q^{8} - 3 q^{9} + q^{10} - q^{11} + 2 q^{13} - q^{16} - 6 q^{17} - 3 q^{18} + q^{19} + O(q^{20})$$

## 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 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.