↑

Properties

Label 20.20.1323583805...1213.1
Degree $20$
Signature $[20, 0]$
Discriminant $13^{10}\cdot 277^{9}$
Root discriminant $45.30$
Ramified primes $13, 277$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{20}$ (as 20T10)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-103, 1089, 17377, -3681, -90960, 29276, 175998, -78936, -165018, 92609, 79884, -55009, -18778, 17203, 1437, -2755, 139, 210, -26, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 26*x^18 + 210*x^17 + 139*x^16 - 2755*x^15 + 1437*x^14 + 17203*x^13 - 18778*x^12 - 55009*x^11 + 79884*x^10 + 92609*x^9 - 165018*x^8 - 78936*x^7 + 175998*x^6 + 29276*x^5 - 90960*x^4 - 3681*x^3 + 17377*x^2 + 1089*x - 103)
 
gp: K = bnfinit(x^20 - 6*x^19 - 26*x^18 + 210*x^17 + 139*x^16 - 2755*x^15 + 1437*x^14 + 17203*x^13 - 18778*x^12 - 55009*x^11 + 79884*x^10 + 92609*x^9 - 165018*x^8 - 78936*x^7 + 175998*x^6 + 29276*x^5 - 90960*x^4 - 3681*x^3 + 17377*x^2 + 1089*x - 103, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 26 x^{18} + 210 x^{17} + 139 x^{16} - 2755 x^{15} + 1437 x^{14} + 17203 x^{13} - 18778 x^{12} - 55009 x^{11} + 79884 x^{10} + 92609 x^{9} - 165018 x^{8} - 78936 x^{7} + 175998 x^{6} + 29276 x^{5} - 90960 x^{4} - 3681 x^{3} + 17377 x^{2} + 1089 x - 103 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1323583805102048234846935217701213=13^{10}\cdot 277^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.30$
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:  $13, 277$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{287} a^{18} + \frac{22}{287} a^{17} + \frac{73}{287} a^{16} + \frac{64}{287} a^{15} + \frac{65}{287} a^{14} + \frac{130}{287} a^{13} - \frac{115}{287} a^{12} - \frac{132}{287} a^{11} - \frac{6}{41} a^{10} + \frac{5}{287} a^{9} + \frac{20}{41} a^{8} + \frac{95}{287} a^{7} + \frac{4}{41} a^{6} - \frac{18}{41} a^{5} - \frac{143}{287} a^{4} + \frac{9}{287} a^{3} - \frac{131}{287} a^{2} + \frac{52}{287} a - \frac{114}{287}$, $\frac{1}{78238268504079345540044213999} a^{19} + \frac{5810782480621308633522823}{78238268504079345540044213999} a^{18} + \frac{18128702565955549466261012255}{78238268504079345540044213999} a^{17} - \frac{329770730363697782113742357}{78238268504079345540044213999} a^{16} + \frac{32920748309984106723602416780}{78238268504079345540044213999} a^{15} - \frac{4207191047390147476829387058}{78238268504079345540044213999} a^{14} + \frac{18591023865195911185887484004}{78238268504079345540044213999} a^{13} - \frac{2108340496380002777932597959}{78238268504079345540044213999} a^{12} - \frac{17719233308782975081810404811}{78238268504079345540044213999} a^{11} - \frac{25611054796503380608862083019}{78238268504079345540044213999} a^{10} + \frac{16184651392179731416990690738}{78238268504079345540044213999} a^{9} + \frac{21946391798512914074299227757}{78238268504079345540044213999} a^{8} - \frac{7816314397777856751507326113}{78238268504079345540044213999} a^{7} - \frac{102319619234771438771140040}{1596699357226109092653963551} a^{6} + \frac{871354752224320757129707687}{1908250451319008427805956439} a^{5} + \frac{5614219940867854285106239528}{78238268504079345540044213999} a^{4} - \frac{33322329303397137014952067627}{78238268504079345540044213999} a^{3} + \frac{6691437251082261232145512493}{78238268504079345540044213999} a^{2} + \frac{33344606731844687438434336027}{78238268504079345540044213999} a - \frac{188835434250186107356488683}{759594839845430539223730233}$

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:  $19$
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:  \( 12870125339.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 13 conjugacy class representatives for $D_{20}$
Character table for $D_{20}$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.46813.1, 5.5.12967201.1, 10.10.2185927923067213.1

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: 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 $20$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ $20$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
277Data not computed