# Properties

 Label 20.2.558...771.1 Degree $20$ Signature $[2, 9]$ Discriminant $-5.589\times 10^{32}$ Root discriminant $43.39$ Ramified primes $7, 19$ Class number $1$ (GRH) Class group trivial (GRH) Galois group $D_{20}$ (as 20T10)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 38*x^18 - 133*x^17 + 323*x^16 - 570*x^15 + 608*x^14 + 608*x^13 - 2280*x^12 + 5662*x^11 - 2527*x^10 + 4731*x^9 + 8265*x^8 - 6935*x^7 + 27265*x^6 - 20539*x^5 + 29469*x^4 - 20178*x^3 + 16226*x^2 - 6696*x + 729)

gp: K = bnfinit(x^20 - 8*x^19 + 38*x^18 - 133*x^17 + 323*x^16 - 570*x^15 + 608*x^14 + 608*x^13 - 2280*x^12 + 5662*x^11 - 2527*x^10 + 4731*x^9 + 8265*x^8 - 6935*x^7 + 27265*x^6 - 20539*x^5 + 29469*x^4 - 20178*x^3 + 16226*x^2 - 6696*x + 729, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![729, -6696, 16226, -20178, 29469, -20539, 27265, -6935, 8265, 4731, -2527, 5662, -2280, 608, 608, -570, 323, -133, 38, -8, 1]);

$$x^{20} - 8 x^{19} + 38 x^{18} - 133 x^{17} + 323 x^{16} - 570 x^{15} + 608 x^{14} + 608 x^{13} - 2280 x^{12} + 5662 x^{11} - 2527 x^{10} + 4731 x^{9} + 8265 x^{8} - 6935 x^{7} + 27265 x^{6} - 20539 x^{5} + 29469 x^{4} - 20178 x^{3} + 16226 x^{2} - 6696 x + 729$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $20$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[2, 9]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-558854584859141340505879659895771$$$$\medspace = -\,7^{10}\cdot 19^{19}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $43.39$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $7, 19$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $2$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{10} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{1}{3} a^{6} + \frac{1}{9} a^{5} + \frac{1}{3} a^{4} + \frac{2}{9} a^{3} - \frac{2}{9} a^{2} + \frac{2}{9} a$, $\frac{1}{45} a^{13} - \frac{2}{45} a^{12} - \frac{1}{45} a^{11} - \frac{7}{45} a^{10} + \frac{2}{45} a^{9} - \frac{2}{15} a^{8} + \frac{1}{45} a^{7} + \frac{22}{45} a^{6} + \frac{13}{45} a^{5} - \frac{4}{45} a^{4} - \frac{7}{15} a^{3} + \frac{2}{5} a^{2} + \frac{8}{45} a - \frac{2}{5}$, $\frac{1}{45} a^{14} + \frac{2}{15} a^{11} - \frac{2}{45} a^{10} - \frac{2}{45} a^{9} - \frac{1}{45} a^{8} - \frac{1}{45} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{16}{45} a^{4} + \frac{1}{45} a^{3} - \frac{11}{45} a^{2} + \frac{8}{45} a + \frac{1}{5}$, $\frac{1}{135} a^{15} + \frac{1}{135} a^{14} + \frac{2}{45} a^{12} - \frac{11}{135} a^{11} - \frac{19}{135} a^{10} - \frac{2}{15} a^{9} - \frac{2}{135} a^{8} - \frac{4}{135} a^{7} - \frac{7}{45} a^{6} - \frac{62}{135} a^{5} + \frac{62}{135} a^{4} - \frac{11}{27} a^{3} - \frac{1}{45} a^{2} - \frac{43}{135} a + \frac{2}{5}$, $\frac{1}{135} a^{16} - \frac{1}{135} a^{14} - \frac{1}{27} a^{12} - \frac{2}{135} a^{11} - \frac{2}{135} a^{10} + \frac{4}{135} a^{9} - \frac{11}{135} a^{8} + \frac{22}{135} a^{7} - \frac{38}{135} a^{6} - \frac{44}{135} a^{5} - \frac{16}{45} a^{4} - \frac{2}{135} a^{3} + \frac{32}{135} a^{2} + \frac{4}{135} a + \frac{2}{5}$, $\frac{1}{135} a^{17} + \frac{1}{135} a^{14} + \frac{1}{135} a^{13} + \frac{7}{135} a^{12} - \frac{19}{135} a^{11} + \frac{2}{15} a^{10} - \frac{17}{135} a^{9} + \frac{14}{135} a^{8} - \frac{7}{45} a^{7} + \frac{67}{135} a^{6} + \frac{28}{135} a^{5} + \frac{4}{15} a^{4} + \frac{16}{135} a^{3} - \frac{56}{135} a^{2} + \frac{44}{135} a - \frac{2}{5}$, $\frac{1}{2025} a^{18} + \frac{7}{2025} a^{17} - \frac{1}{405} a^{16} + \frac{1}{2025} a^{15} - \frac{17}{2025} a^{14} + \frac{2}{2025} a^{13} + \frac{34}{2025} a^{12} - \frac{61}{675} a^{11} + \frac{308}{2025} a^{10} - \frac{134}{2025} a^{9} + \frac{11}{225} a^{8} + \frac{98}{2025} a^{7} + \frac{2}{225} a^{6} - \frac{919}{2025} a^{5} + \frac{796}{2025} a^{4} + \frac{37}{225} a^{3} + \frac{776}{2025} a^{2} + \frac{26}{675} a + \frac{8}{25}$, $\frac{1}{1486299344023045856654884219725} a^{19} + \frac{19799292315508951344617354}{87429373177826226862052012925} a^{18} + \frac{73355763810520263497339284}{114330718771003527434991093825} a^{17} - \frac{5337429646775956199944866824}{1486299344023045856654884219725} a^{16} - \frac{4657498925220160484004727676}{1486299344023045856654884219725} a^{15} + \frac{913188289314214809706021723}{297259868804609171330976843945} a^{14} - \frac{3797741184436848776229886139}{1486299344023045856654884219725} a^{13} - \frac{85936255733656706112123302}{12703413196778169714999010425} a^{12} - \frac{40229454640718049499902381026}{297259868804609171330976843945} a^{11} + \frac{67045699865285811184058958229}{1486299344023045856654884219725} a^{10} + \frac{664646400220409725397642332}{19817324586973944755398456263} a^{9} + \frac{62302299384296487929898445262}{1486299344023045856654884219725} a^{8} + \frac{12912145933134306396767316257}{495433114674348618884961406575} a^{7} - \frac{23694780683256160578769307242}{114330718771003527434991093825} a^{6} + \frac{18001307206740566375813567396}{87429373177826226862052012925} a^{5} + \frac{229983918148422167554682139298}{495433114674348618884961406575} a^{4} - \frac{129816578006258372469643589131}{1486299344023045856654884219725} a^{3} + \frac{3219072564131263512561634288}{495433114674348618884961406575} a^{2} - \frac{68423867988464735805928786463}{495433114674348618884961406575} a + \frac{1769269393939747239258651038}{18349374617568467366109681725}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $10$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$1336347065.64$$ (assuming GRH) sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{9}\cdot 1336347065.64 \cdot 1}{2\sqrt{558854584859141340505879659895771}}\approx 1.72551471051$ (assuming GRH)

## Galois group

$D_{20}$ (as 20T10):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 40 The 13 conjugacy class representatives for $D_{20}$ Character table for $D_{20}$

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Sibling fields

 Galois closure: data not computed Degree 20 sibling: Deg 20

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type $20$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

gp: idealfactors = idealprimedec(K, p); \\ get the data

gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

magma: idealfactors := Factorization(p*Integers(K)); // get the data

magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 7.4.2.1x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 7.4.2.1x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19Data not computed

This field is associated with the points of order 19 on an elliptic curve in the isogeny class 49.a, which has complex multiplication by $\Q(\sqrt{-7})$.