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Properties

Label 20.6.16405056404...2992.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{20}\cdot 11^{16}\cdot 23^{7}$
Root discriminant $40.81$
Ramified primes $2, 11, 23$
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![263, 2748, -4063, -27680, 54384, 52442, -50153, 75706, -42832, 9816, 13387, -21060, 17055, -10874, 5300, -2218, 747, -212, 48, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 48*x^18 - 212*x^17 + 747*x^16 - 2218*x^15 + 5300*x^14 - 10874*x^13 + 17055*x^12 - 21060*x^11 + 13387*x^10 + 9816*x^9 - 42832*x^8 + 75706*x^7 - 50153*x^6 + 52442*x^5 + 54384*x^4 - 27680*x^3 - 4063*x^2 + 2748*x + 263)
 
gp: K = bnfinit(x^20 - 8*x^19 + 48*x^18 - 212*x^17 + 747*x^16 - 2218*x^15 + 5300*x^14 - 10874*x^13 + 17055*x^12 - 21060*x^11 + 13387*x^10 + 9816*x^9 - 42832*x^8 + 75706*x^7 - 50153*x^6 + 52442*x^5 + 54384*x^4 - 27680*x^3 - 4063*x^2 + 2748*x + 263, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 48 x^{18} - 212 x^{17} + 747 x^{16} - 2218 x^{15} + 5300 x^{14} - 10874 x^{13} + 17055 x^{12} - 21060 x^{11} + 13387 x^{10} + 9816 x^{9} - 42832 x^{8} + 75706 x^{7} - 50153 x^{6} + 52442 x^{5} + 54384 x^{4} - 27680 x^{3} - 4063 x^{2} + 2748 x + 263 \)

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:  \(-164050564045619941441358756052992=-\,2^{20}\cdot 11^{16}\cdot 23^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.81$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{487495330430846874138362309760974970599075710943} a^{19} - \frac{63733367432107276237351989612920072492128847596}{487495330430846874138362309760974970599075710943} a^{18} + \frac{40970251833036556260359440809988494860455565764}{487495330430846874138362309760974970599075710943} a^{17} + \frac{228207802844446680432479663092810228924961069443}{487495330430846874138362309760974970599075710943} a^{16} + \frac{62300558596286828018807923405290598100869363882}{487495330430846874138362309760974970599075710943} a^{15} - \frac{78693917878559026762447926509324685287619062081}{487495330430846874138362309760974970599075710943} a^{14} - \frac{80939532973293959614232467539030007926955350061}{487495330430846874138362309760974970599075710943} a^{13} + \frac{129149547153848249429882535009051870839368851836}{487495330430846874138362309760974970599075710943} a^{12} + \frac{203553034786477299216627258945246000623332942037}{487495330430846874138362309760974970599075710943} a^{11} - \frac{237920841601963934553331342471567255695190505281}{487495330430846874138362309760974970599075710943} a^{10} - \frac{8381243642825531843482507962162985884761787635}{487495330430846874138362309760974970599075710943} a^{9} - \frac{34303628186716946250817410421848359512066403298}{487495330430846874138362309760974970599075710943} a^{8} + \frac{103667617575771739853150318530342023321588933497}{487495330430846874138362309760974970599075710943} a^{7} - \frac{105452486409866160383951019729775166827263427429}{487495330430846874138362309760974970599075710943} a^{6} - \frac{199772761717753612568381371369182135371516307991}{487495330430846874138362309760974970599075710943} a^{5} - \frac{4896986140360784139034222503369717242191906192}{11337100707694113352054937436301743502304086301} a^{4} - \frac{117600072330312316082732394437713801195886844512}{487495330430846874138362309760974970599075710943} a^{3} + \frac{189121266570352233453790585013976957077149732888}{487495330430846874138362309760974970599075710943} a^{2} - \frac{5958397092974346802075745137181275886997145450}{487495330430846874138362309760974970599075710943} a + \frac{71763218858672555198111067389283557778012190683}{487495330430846874138362309760974970599075710943}$

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:  \( 212420669.483 \) (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

\(\Q(\zeta_{11})^+\), 10.10.116117348402176.2

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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$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}$$[\ ]_{5}^{2}$
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$