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Properties

Label 20.6.35028156925...9072.2
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{20}\cdot 11^{16}\cdot 727$
Root discriminant $18.93$
Ramified primes $2, 11, 727$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T749

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 8, -98, 290, -365, 522, -2108, 5836, -9191, 8520, -4043, -160, 1303, -380, -436, 446, -156, -4, 23, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 23*x^18 - 4*x^17 - 156*x^16 + 446*x^15 - 436*x^14 - 380*x^13 + 1303*x^12 - 160*x^11 - 4043*x^10 + 8520*x^9 - 9191*x^8 + 5836*x^7 - 2108*x^6 + 522*x^5 - 365*x^4 + 290*x^3 - 98*x^2 + 8*x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 23*x^18 - 4*x^17 - 156*x^16 + 446*x^15 - 436*x^14 - 380*x^13 + 1303*x^12 - 160*x^11 - 4043*x^10 + 8520*x^9 - 9191*x^8 + 5836*x^7 - 2108*x^6 + 522*x^5 - 365*x^4 + 290*x^3 - 98*x^2 + 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 23 x^{18} - 4 x^{17} - 156 x^{16} + 446 x^{15} - 436 x^{14} - 380 x^{13} + 1303 x^{12} - 160 x^{11} - 4043 x^{10} + 8520 x^{9} - 9191 x^{8} + 5836 x^{7} - 2108 x^{6} + 522 x^{5} - 365 x^{4} + 290 x^{3} - 98 x^{2} + 8 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-35028156925416005746819072=-\,2^{20}\cdot 11^{16}\cdot 727\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.93$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 11, 727$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{20131174322029} a^{19} + \frac{718372241719}{20131174322029} a^{18} + \frac{892922312468}{20131174322029} a^{17} + \frac{4026285170113}{20131174322029} a^{16} - \frac{4304702682012}{20131174322029} a^{15} - \frac{9071495571787}{20131174322029} a^{14} - \frac{2861981399298}{20131174322029} a^{13} - \frac{6192532853426}{20131174322029} a^{12} + \frac{8381578997708}{20131174322029} a^{11} - \frac{9119853642770}{20131174322029} a^{10} + \frac{861615139553}{20131174322029} a^{9} + \frac{6269250566938}{20131174322029} a^{8} + \frac{6585411440086}{20131174322029} a^{7} - \frac{7677550683565}{20131174322029} a^{6} - \frac{3178124950271}{20131174322029} a^{5} - \frac{5682780713536}{20131174322029} a^{4} - \frac{5521324827328}{20131174322029} a^{3} + \frac{9419801213624}{20131174322029} a^{2} - \frac{869815178535}{20131174322029} a + \frac{1497595394094}{20131174322029}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

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

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $12$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 99224.2125593 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T749:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n749 are not computed
Character table for t20n749 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.219503494144.2

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

Sibling fields

Degree 20 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.10.10.1$x^{10} - 9 x^{8} + 54 x^{6} - 38 x^{4} + 41 x^{2} - 17$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
2.10.10.1$x^{10} - 9 x^{8} + 54 x^{6} - 38 x^{4} + 41 x^{2} - 17$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
727Data not computed