# Properties

 Label 28224by Number of curves $6$ Conductor $28224$ CM no Rank $1$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("by1")

sage: E.isogeny_class()

## Elliptic curves in class 28224by

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
28224.fl5 28224by1 $$[0, 0, 0, 1176, 19208]$$ $$2048/3$$ $$-263473523712$$ $$[2]$$ $$24576$$ $$0.87691$$ $$\Gamma_0(N)$$-optimal
28224.fl4 28224by2 $$[0, 0, 0, -7644, 192080]$$ $$35152/9$$ $$12646729138176$$ $$[2, 2]$$ $$49152$$ $$1.2235$$
28224.fl3 28224by3 $$[0, 0, 0, -42924, -3265360]$$ $$1556068/81$$ $$455282248974336$$ $$[2, 2]$$ $$98304$$ $$1.5701$$
28224.fl2 28224by4 $$[0, 0, 0, -113484, 14713328]$$ $$28756228/3$$ $$16862305517568$$ $$[2]$$ $$98304$$ $$1.5701$$
28224.fl6 28224by5 $$[0, 0, 0, 27636, -12946192]$$ $$207646/6561$$ $$-73755724333842432$$ $$[2]$$ $$196608$$ $$1.9166$$
28224.fl1 28224by6 $$[0, 0, 0, -677964, -214860688]$$ $$3065617154/9$$ $$101173833105408$$ $$[2]$$ $$196608$$ $$1.9166$$

## Rank

sage: E.rank()

The elliptic curves in class 28224by have rank $$1$$.

## Complex multiplication

The elliptic curves in class 28224by do not have complex multiplication.

## Modular form 28224.2.a.by

sage: E.q_eigenform(10)

$$q + 2q^{5} + 4q^{11} - 2q^{13} + 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.