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Properties

Label 20.0.28016810838...1888.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{34}\cdot 277^{5}$
Root discriminant $13.25$
Ramified primes $2, 277$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T797

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -8, 20, -32, 12, 84, -204, 184, 30, -240, 276, -248, 314, -404, 380, -262, 148, -74, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 30*x^18 - 74*x^17 + 148*x^16 - 262*x^15 + 380*x^14 - 404*x^13 + 314*x^12 - 248*x^11 + 276*x^10 - 240*x^9 + 30*x^8 + 184*x^7 - 204*x^6 + 84*x^5 + 12*x^4 - 32*x^3 + 20*x^2 - 8*x + 2)
 
gp: K = bnfinit(x^20 - 8*x^19 + 30*x^18 - 74*x^17 + 148*x^16 - 262*x^15 + 380*x^14 - 404*x^13 + 314*x^12 - 248*x^11 + 276*x^10 - 240*x^9 + 30*x^8 + 184*x^7 - 204*x^6 + 84*x^5 + 12*x^4 - 32*x^3 + 20*x^2 - 8*x + 2, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 30 x^{18} - 74 x^{17} + 148 x^{16} - 262 x^{15} + 380 x^{14} - 404 x^{13} + 314 x^{12} - 248 x^{11} + 276 x^{10} - 240 x^{9} + 30 x^{8} + 184 x^{7} - 204 x^{6} + 84 x^{5} + 12 x^{4} - 32 x^{3} + 20 x^{2} - 8 x + 2 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(28016810838376881061888=2^{34}\cdot 277^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.25$
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:  $2, 277$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{138963161} a^{19} - \frac{3311402}{138963161} a^{18} + \frac{51606230}{138963161} a^{17} - \frac{2776554}{138963161} a^{16} + \frac{44635181}{138963161} a^{15} + \frac{21566049}{138963161} a^{14} + \frac{39028618}{138963161} a^{13} - \frac{56628032}{138963161} a^{12} + \frac{2164556}{138963161} a^{11} + \frac{22093068}{138963161} a^{10} - \frac{29150134}{138963161} a^{9} + \frac{15190609}{138963161} a^{8} - \frac{67516975}{138963161} a^{7} - \frac{22482668}{138963161} a^{6} + \frac{14265882}{138963161} a^{5} + \frac{14669882}{138963161} a^{4} + \frac{9844757}{138963161} a^{3} + \frac{54530344}{138963161} a^{2} + \frac{56727104}{138963161} a + \frac{23432503}{138963161}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{730324}{474277} a^{19} - \frac{4812224}{474277} a^{18} + \frac{14157385}{474277} a^{17} - \frac{27412583}{474277} a^{16} + \frac{48474845}{474277} a^{15} - \frac{79271373}{474277} a^{14} + \frac{85352698}{474277} a^{13} - \frac{39513290}{474277} a^{12} + \frac{8836120}{474277} a^{11} - \frac{49050069}{474277} a^{10} + \frac{65425530}{474277} a^{9} + \frac{14933262}{474277} a^{8} - \frac{82882834}{474277} a^{7} + \frac{51140218}{474277} a^{6} + \frac{7551648}{474277} a^{5} - \frac{19627780}{474277} a^{4} + \frac{7959298}{474277} a^{3} - \frac{1622570}{474277} a^{2} - \frac{417216}{474277} a + \frac{826775}{474277} \) (order $4$)
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:  \( 2417.97246894 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n797 are not computed
Character table for t20n797 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.1.4432.1, 10.0.1257127936.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
277Data not computed