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Properties

Label 20.2.35786702638...0000.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{16}\cdot 5^{11}\cdot 19^{5}\cdot 461^{4}$
Root discriminant $30.04$
Ramified primes $2, 5, 19, 461$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1023

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20, 240, 1084, 2248, 2092, 1010, 642, 816, 3158, 722, 2183, -1335, 1783, -1376, 891, -497, 219, -80, 25, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 25*x^18 - 80*x^17 + 219*x^16 - 497*x^15 + 891*x^14 - 1376*x^13 + 1783*x^12 - 1335*x^11 + 2183*x^10 + 722*x^9 + 3158*x^8 + 816*x^7 + 642*x^6 + 1010*x^5 + 2092*x^4 + 2248*x^3 + 1084*x^2 + 240*x + 20)
 
gp: K = bnfinit(x^20 - 5*x^19 + 25*x^18 - 80*x^17 + 219*x^16 - 497*x^15 + 891*x^14 - 1376*x^13 + 1783*x^12 - 1335*x^11 + 2183*x^10 + 722*x^9 + 3158*x^8 + 816*x^7 + 642*x^6 + 1010*x^5 + 2092*x^4 + 2248*x^3 + 1084*x^2 + 240*x + 20, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 25 x^{18} - 80 x^{17} + 219 x^{16} - 497 x^{15} + 891 x^{14} - 1376 x^{13} + 1783 x^{12} - 1335 x^{11} + 2183 x^{10} + 722 x^{9} + 3158 x^{8} + 816 x^{7} + 642 x^{6} + 1010 x^{5} + 2092 x^{4} + 2248 x^{3} + 1084 x^{2} + 240 x + 20 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-357867026381710908800000000000=-\,2^{16}\cdot 5^{11}\cdot 19^{5}\cdot 461^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.04$
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:  $2, 5, 19, 461$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{6} a^{16} - \frac{1}{6} a^{15} - \frac{1}{2} a^{14} + \frac{1}{3} a^{13} - \frac{1}{6} a^{12} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{17} - \frac{1}{6} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{2} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{114} a^{18} - \frac{3}{38} a^{17} + \frac{13}{114} a^{15} + \frac{17}{57} a^{14} - \frac{13}{38} a^{13} + \frac{28}{57} a^{12} - \frac{1}{2} a^{11} - \frac{3}{19} a^{10} - \frac{47}{114} a^{9} + \frac{5}{19} a^{8} + \frac{49}{114} a^{7} + \frac{11}{114} a^{6} - \frac{16}{57} a^{5} + \frac{2}{57} a^{4} - \frac{3}{19} a^{3} - \frac{3}{19} a^{2} + \frac{6}{19} a - \frac{7}{57}$, $\frac{1}{2749480639891272974375351490} a^{19} + \frac{7616896160516020179546313}{2749480639891272974375351490} a^{18} - \frac{71030814749445540211314837}{916493546630424324791783830} a^{17} - \frac{129513522929667568928663573}{2749480639891272974375351490} a^{16} + \frac{6018145115913017233312361}{274948063989127297437535149} a^{15} - \frac{425838267373936100173317976}{1374740319945636487187675745} a^{14} + \frac{48096714895043084000496871}{549896127978254594875070298} a^{13} - \frac{589720971037245923354013301}{2749480639891272974375351490} a^{12} - \frac{39441368626902903024164196}{91649354663042432479178383} a^{11} + \frac{43342411925423535275600122}{274948063989127297437535149} a^{10} + \frac{129407146721816500950975523}{2749480639891272974375351490} a^{9} + \frac{740277634681229679753192331}{2749480639891272974375351490} a^{8} + \frac{36738657137974228671575397}{916493546630424324791783830} a^{7} - \frac{202199084874913594328556677}{916493546630424324791783830} a^{6} + \frac{88349651698175312389666342}{1374740319945636487187675745} a^{5} + \frac{111592387446741044066240816}{1374740319945636487187675745} a^{4} - \frac{1032568039502517326915722}{458246773315212162395891915} a^{3} - \frac{10110045093930819774642832}{24118251227116429599783785} a^{2} + \frac{62897616103131658560741058}{274948063989127297437535149} a + \frac{63648744922183311101444152}{274948063989127297437535149}$

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:  $10$
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:  \( 18580459.6356 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1023:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 324 conjugacy class representatives for t20n1023 are not computed
Character table for t20n1023 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.61376064800000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $20$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.10.0.1$x^{10} + x^{2} - 2 x + 14$$1$$10$$0$$C_{10}$$[\ ]^{10}$
461Data not computed