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Properties

Label 20.6.33974138629...4375.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,5^{14}\cdot 19^{9}\cdot 29^{7}$
Root discriminant $37.72$
Ramified primes $5, 19, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T955

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -8, 51, 23, 253, -144, 428, -723, -1466, 2170, -2812, 560, 2137, -845, 689, -1, 6, -14, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 21*x^18 - 14*x^17 + 6*x^16 - x^15 + 689*x^14 - 845*x^13 + 2137*x^12 + 560*x^11 - 2812*x^10 + 2170*x^9 - 1466*x^8 - 723*x^7 + 428*x^6 - 144*x^5 + 253*x^4 + 23*x^3 + 51*x^2 - 8*x - 1)
 
gp: K = bnfinit(x^20 - 6*x^19 + 21*x^18 - 14*x^17 + 6*x^16 - x^15 + 689*x^14 - 845*x^13 + 2137*x^12 + 560*x^11 - 2812*x^10 + 2170*x^9 - 1466*x^8 - 723*x^7 + 428*x^6 - 144*x^5 + 253*x^4 + 23*x^3 + 51*x^2 - 8*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 21 x^{18} - 14 x^{17} + 6 x^{16} - x^{15} + 689 x^{14} - 845 x^{13} + 2137 x^{12} + 560 x^{11} - 2812 x^{10} + 2170 x^{9} - 1466 x^{8} - 723 x^{7} + 428 x^{6} - 144 x^{5} + 253 x^{4} + 23 x^{3} + 51 x^{2} - 8 x - 1 \)

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:  \(-33974138629905542100286865234375=-\,5^{14}\cdot 19^{9}\cdot 29^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.72$
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:  $5, 19, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{72950097057444072842743091151391022} a^{19} - \frac{4238978697916512446370134557856702}{36475048528722036421371545575695511} a^{18} - \frac{384271277368993176944882761671781}{36475048528722036421371545575695511} a^{17} - \frac{1578375386860666401103083982011011}{72950097057444072842743091151391022} a^{16} + \frac{10190807157531641563695964965839847}{72950097057444072842743091151391022} a^{15} + \frac{26029494951242198606405451502274207}{72950097057444072842743091151391022} a^{14} - \frac{4155674995213667799971832145115236}{36475048528722036421371545575695511} a^{13} - \frac{254369971884972735989936489948447}{72950097057444072842743091151391022} a^{12} - \frac{5737704580237593781341002318642335}{72950097057444072842743091151391022} a^{11} - \frac{7607362819760891456601961998683467}{72950097057444072842743091151391022} a^{10} - \frac{11870208547292734842430594083922857}{36475048528722036421371545575695511} a^{9} - \frac{5013662612684786363185384606822923}{72950097057444072842743091151391022} a^{8} - \frac{29641699895882795261398600936784345}{72950097057444072842743091151391022} a^{7} + \frac{15103682690769542795289830387048955}{72950097057444072842743091151391022} a^{6} - \frac{12233723671570813278341914232988711}{72950097057444072842743091151391022} a^{5} + \frac{9183804351944323222687839582347955}{36475048528722036421371545575695511} a^{4} + \frac{2512029056248148671934933550609488}{36475048528722036421371545575695511} a^{3} + \frac{17872429373497972509518096315186890}{36475048528722036421371545575695511} a^{2} + \frac{8826764425432792158402998503192363}{36475048528722036421371545575695511} a + \frac{22101118915960294779893381633308825}{72950097057444072842743091151391022}$

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:  $12$
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:  \( 110224914.643 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T955:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 819200
The 275 conjugacy class representatives for t20n955 are not computed
Character table for t20n955 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.9932496465625.1

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
Degree 40 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 $16{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ $16{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ $16{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ $20$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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

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
5Data not computed
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.8.0.1$x^{8} - x + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$