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Properties

Label 20.4.11982070107...2437.1
Degree $20$
Signature $[4, 8]$
Discriminant $13^{11}\cdot 401^{8}$
Root discriminant $45.07$
Ramified primes $13, 401$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2873, 24167, -81549, 165620, -223144, 198081, -87282, -60450, 170528, -210750, 185448, -130718, 75992, -37276, 15470, -5416, 1595, -381, 74, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 74*x^18 - 381*x^17 + 1595*x^16 - 5416*x^15 + 15470*x^14 - 37276*x^13 + 75992*x^12 - 130718*x^11 + 185448*x^10 - 210750*x^9 + 170528*x^8 - 60450*x^7 - 87282*x^6 + 198081*x^5 - 223144*x^4 + 165620*x^3 - 81549*x^2 + 24167*x - 2873)
 
gp: K = bnfinit(x^20 - 10*x^19 + 74*x^18 - 381*x^17 + 1595*x^16 - 5416*x^15 + 15470*x^14 - 37276*x^13 + 75992*x^12 - 130718*x^11 + 185448*x^10 - 210750*x^9 + 170528*x^8 - 60450*x^7 - 87282*x^6 + 198081*x^5 - 223144*x^4 + 165620*x^3 - 81549*x^2 + 24167*x - 2873, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 74 x^{18} - 381 x^{17} + 1595 x^{16} - 5416 x^{15} + 15470 x^{14} - 37276 x^{13} + 75992 x^{12} - 130718 x^{11} + 185448 x^{10} - 210750 x^{9} + 170528 x^{8} - 60450 x^{7} - 87282 x^{6} + 198081 x^{5} - 223144 x^{4} + 165620 x^{3} - 81549 x^{2} + 24167 x - 2873 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1198207010758520322015083315072437=13^{11}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.07$
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:  $13, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{117} a^{16} - \frac{8}{117} a^{15} - \frac{8}{117} a^{14} - \frac{38}{117} a^{13} - \frac{20}{117} a^{12} + \frac{16}{117} a^{11} + \frac{4}{9} a^{10} - \frac{47}{117} a^{9} - \frac{35}{117} a^{8} + \frac{4}{39} a^{6} - \frac{1}{13} a^{5} + \frac{14}{39} a^{4} - \frac{8}{39} a^{3} + \frac{3}{13} a^{2} + \frac{1}{3} a - \frac{4}{9}$, $\frac{1}{117} a^{17} + \frac{2}{39} a^{15} + \frac{5}{39} a^{14} - \frac{4}{39} a^{13} - \frac{3}{13} a^{12} + \frac{8}{39} a^{11} + \frac{2}{13} a^{10} + \frac{2}{13} a^{9} - \frac{46}{117} a^{8} - \frac{3}{13} a^{7} + \frac{1}{13} a^{6} + \frac{16}{39} a^{5} + \frac{10}{39} a^{3} - \frac{19}{39} a^{2} - \frac{1}{9} a - \frac{2}{9}$, $\frac{1}{232942671} a^{18} - \frac{1}{25882519} a^{17} + \frac{67895}{17918667} a^{16} - \frac{7060876}{232942671} a^{15} + \frac{9866030}{232942671} a^{14} + \frac{54502406}{232942671} a^{13} - \frac{33731155}{232942671} a^{12} + \frac{104359130}{232942671} a^{11} + \frac{38474201}{232942671} a^{10} - \frac{18026831}{232942671} a^{9} + \frac{52818479}{232942671} a^{8} + \frac{32930774}{77647557} a^{7} - \frac{11642705}{77647557} a^{6} - \frac{11981446}{77647557} a^{5} + \frac{4373834}{25882519} a^{4} + \frac{3601933}{25882519} a^{3} + \frac{793397}{17918667} a^{2} - \frac{6089021}{17918667} a - \frac{471596}{1378359}$, $\frac{1}{394371942003} a^{19} + \frac{93}{43819104667} a^{18} + \frac{1386585269}{394371942003} a^{17} - \frac{926787697}{394371942003} a^{16} - \frac{61188966760}{394371942003} a^{15} - \frac{29460979585}{394371942003} a^{14} - \frac{14615220808}{394371942003} a^{13} - \frac{152716072312}{394371942003} a^{12} + \frac{22009311614}{394371942003} a^{11} - \frac{93472919129}{394371942003} a^{10} - \frac{71731274932}{394371942003} a^{9} - \frac{36960735854}{131457314001} a^{8} - \frac{2870776028}{131457314001} a^{7} - \frac{42342280258}{131457314001} a^{6} + \frac{7546038117}{43819104667} a^{5} - \frac{7256385818}{43819104667} a^{4} - \frac{129765033568}{394371942003} a^{3} - \frac{10072291769}{30336303231} a^{2} - \frac{2977338029}{30336303231} a + \frac{149726674}{777853929}$

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:  $11$
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:  \( 347743451.644 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.5.160801.1, 10.10.9600508843720093.1

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 $20$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ $20$ $20$ $20$ R ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ $20$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
401Data not computed