# Properties

 Label 28224.bd Number of curves $4$ Conductor $28224$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("bd1")

sage: E.isogeny_class()

## Elliptic curves in class 28224.bd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
28224.bd1 28224ck4 $$[0, 0, 0, -527436, 147435120]$$ $$1443468546/7$$ $$78690759081984$$ $$$$ $$196608$$ $$1.8668$$
28224.bd2 28224ck3 $$[0, 0, 0, -104076, -10224144]$$ $$11090466/2401$$ $$26990930365120512$$ $$$$ $$196608$$ $$1.8668$$
28224.bd3 28224ck2 $$[0, 0, 0, -33516, 2222640]$$ $$740772/49$$ $$275417656786944$$ $$[2, 2]$$ $$98304$$ $$1.5202$$
28224.bd4 28224ck1 $$[0, 0, 0, 1764, 148176]$$ $$432/7$$ $$-9836344885248$$ $$$$ $$49152$$ $$1.1737$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 28224.2.a.bd

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 