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Properties

Label 20.2.33313079744...9872.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{10}\cdot 11^{17}\cdot 23^{5}$
Root discriminant $23.78$
Ramified primes $2, 11, 23$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T749

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-197, 1562, -5555, 11921, -17674, 19250, -15249, 7783, -707, -3241, 3761, -2430, 877, -4, -247, 171, -60, 1, 10, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 10*x^18 + x^17 - 60*x^16 + 171*x^15 - 247*x^14 - 4*x^13 + 877*x^12 - 2430*x^11 + 3761*x^10 - 3241*x^9 - 707*x^8 + 7783*x^7 - 15249*x^6 + 19250*x^5 - 17674*x^4 + 11921*x^3 - 5555*x^2 + 1562*x - 197)
 
gp: K = bnfinit(x^20 - 5*x^19 + 10*x^18 + x^17 - 60*x^16 + 171*x^15 - 247*x^14 - 4*x^13 + 877*x^12 - 2430*x^11 + 3761*x^10 - 3241*x^9 - 707*x^8 + 7783*x^7 - 15249*x^6 + 19250*x^5 - 17674*x^4 + 11921*x^3 - 5555*x^2 + 1562*x - 197, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 10 x^{18} + x^{17} - 60 x^{16} + 171 x^{15} - 247 x^{14} - 4 x^{13} + 877 x^{12} - 2430 x^{11} + 3761 x^{10} - 3241 x^{9} - 707 x^{8} + 7783 x^{7} - 15249 x^{6} + 19250 x^{5} - 17674 x^{4} + 11921 x^{3} - 5555 x^{2} + 1562 x - 197 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-3331307974402283487149519872=-\,2^{10}\cdot 11^{17}\cdot 23^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.78$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{8} a^{10} - \frac{3}{8} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{15} - \frac{1}{8} a^{14} - \frac{1}{4} a^{13} + \frac{1}{8} a^{12} + \frac{1}{8} a^{11} - \frac{1}{2} a^{10} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{8} a^{5} + \frac{1}{4} a^{4} + \frac{1}{8} a^{2} - \frac{3}{8} a - \frac{1}{8}$, $\frac{1}{176} a^{18} + \frac{7}{176} a^{17} + \frac{3}{176} a^{16} + \frac{1}{11} a^{15} - \frac{9}{176} a^{14} - \frac{29}{176} a^{13} - \frac{5}{22} a^{12} + \frac{87}{176} a^{11} + \frac{61}{176} a^{10} + \frac{65}{176} a^{9} - \frac{43}{88} a^{8} + \frac{4}{11} a^{7} - \frac{3}{16} a^{6} + \frac{67}{176} a^{5} + \frac{43}{88} a^{4} - \frac{71}{176} a^{3} + \frac{1}{44} a^{2} + \frac{41}{88} a - \frac{51}{176}$, $\frac{1}{379100458009272780176} a^{19} - \frac{874874485045652115}{379100458009272780176} a^{18} + \frac{12698298810127322101}{379100458009272780176} a^{17} + \frac{219661957163438853}{17231839000421490008} a^{16} - \frac{11997840782542936793}{379100458009272780176} a^{15} + \frac{21040396367905939681}{379100458009272780176} a^{14} - \frac{34297701110303995171}{189550229004636390088} a^{13} + \frac{70422526817482902531}{379100458009272780176} a^{12} + \frac{60131629905897955431}{379100458009272780176} a^{11} + \frac{9887511870837662401}{34463678000842980016} a^{10} + \frac{65055007846070603}{2153979875052686251} a^{9} + \frac{2778190467107620777}{47387557251159097522} a^{8} + \frac{72278497234016558635}{379100458009272780176} a^{7} + \frac{124933420159585489233}{379100458009272780176} a^{6} + \frac{3892373318076192137}{8615919500210745004} a^{5} + \frac{189305342692115829973}{379100458009272780176} a^{4} + \frac{30381867366459701947}{189550229004636390088} a^{3} + \frac{39436255064863518687}{189550229004636390088} a^{2} + \frac{169296868949093377605}{379100458009272780176} a - \frac{93217911504639210923}{189550229004636390088}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $10$
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:  \( 248857.214333 \) (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

\(\Q(\zeta_{11})^+\), 10.6.113395848049.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.12$x^{10} - 11 x^{8} + 54 x^{6} - 10 x^{4} + 9 x^{2} - 11$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$