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Properties

Label 20.12.1606601353...0000.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 5^{10}\cdot 13^{4}\cdot 29^{6}\cdot 31^{4}$
Root discriminant $40.77$
Ramified primes $2, 5, 13, 29, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1013

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3125, -3125, 13125, 13125, -17875, -20000, 9300, 15800, 750, -6670, -3514, 926, 2098, 432, -668, -248, 135, 51, -19, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 19*x^18 + 51*x^17 + 135*x^16 - 248*x^15 - 668*x^14 + 432*x^13 + 2098*x^12 + 926*x^11 - 3514*x^10 - 6670*x^9 + 750*x^8 + 15800*x^7 + 9300*x^6 - 20000*x^5 - 17875*x^4 + 13125*x^3 + 13125*x^2 - 3125*x - 3125)
 
gp: K = bnfinit(x^20 - 3*x^19 - 19*x^18 + 51*x^17 + 135*x^16 - 248*x^15 - 668*x^14 + 432*x^13 + 2098*x^12 + 926*x^11 - 3514*x^10 - 6670*x^9 + 750*x^8 + 15800*x^7 + 9300*x^6 - 20000*x^5 - 17875*x^4 + 13125*x^3 + 13125*x^2 - 3125*x - 3125, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 19 x^{18} + 51 x^{17} + 135 x^{16} - 248 x^{15} - 668 x^{14} + 432 x^{13} + 2098 x^{12} + 926 x^{11} - 3514 x^{10} - 6670 x^{9} + 750 x^{8} + 15800 x^{7} + 9300 x^{6} - 20000 x^{5} - 17875 x^{4} + 13125 x^{3} + 13125 x^{2} - 3125 x - 3125 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(160660135384776665098240000000000=2^{20}\cdot 5^{10}\cdot 13^{4}\cdot 29^{6}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.77$
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:  $2, 5, 13, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{20} a^{12} + \frac{1}{10} a^{11} + \frac{1}{20} a^{10} - \frac{1}{5} a^{9} - \frac{1}{4} a^{8} + \frac{7}{20} a^{7} - \frac{3}{20} a^{6} + \frac{7}{20} a^{5} - \frac{1}{10} a^{4} + \frac{1}{20} a^{3} - \frac{1}{5} a^{2} - \frac{1}{4} a$, $\frac{1}{20} a^{13} + \frac{1}{10} a^{11} + \frac{1}{5} a^{10} - \frac{1}{10} a^{9} - \frac{3}{20} a^{8} - \frac{1}{10} a^{7} + \frac{2}{5} a^{6} - \frac{1}{20} a^{5} - \frac{1}{2} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{200} a^{14} - \frac{3}{200} a^{13} - \frac{1}{50} a^{12} + \frac{3}{100} a^{11} - \frac{33}{200} a^{9} - \frac{43}{200} a^{8} + \frac{3}{50} a^{7} + \frac{53}{200} a^{6} + \frac{31}{200} a^{5} - \frac{47}{100} a^{4} + \frac{7}{20} a^{3} - \frac{1}{20} a^{2} - \frac{3}{8} a + \frac{1}{8}$, $\frac{1}{200} a^{15} - \frac{3}{200} a^{13} + \frac{1}{50} a^{12} + \frac{1}{25} a^{11} + \frac{17}{200} a^{10} + \frac{6}{25} a^{9} + \frac{3}{200} a^{8} - \frac{11}{200} a^{7} - \frac{1}{20} a^{6} + \frac{9}{200} a^{5} + \frac{9}{100} a^{4} - \frac{1}{4} a^{3} + \frac{17}{40} a^{2} - \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{1000} a^{16} + \frac{1}{500} a^{15} + \frac{1}{1000} a^{14} - \frac{3}{125} a^{13} - \frac{113}{1000} a^{11} - \frac{79}{500} a^{10} - \frac{63}{1000} a^{9} - \frac{247}{1000} a^{8} + \frac{43}{500} a^{7} + \frac{471}{1000} a^{6} - \frac{7}{25} a^{5} + \frac{9}{25} a^{4} - \frac{3}{8} a^{3} - \frac{1}{20} a^{2} + \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{2000} a^{17} + \frac{1}{1000} a^{15} + \frac{1}{500} a^{14} + \frac{43}{2000} a^{13} + \frac{37}{2000} a^{12} - \frac{3}{500} a^{11} + \frac{61}{250} a^{10} - \frac{71}{2000} a^{9} + \frac{51}{400} a^{8} + \frac{327}{1000} a^{7} - \frac{79}{250} a^{6} + \frac{109}{400} a^{5} - \frac{13}{80} a^{4} + \frac{7}{20} a^{3} + \frac{9}{20} a^{2} - \frac{3}{8} a + \frac{5}{16}$, $\frac{1}{10000} a^{18} + \frac{1}{5000} a^{17} - \frac{1}{2500} a^{16} + \frac{1}{625} a^{15} - \frac{1}{2000} a^{14} - \frac{143}{10000} a^{13} + \frac{21}{5000} a^{12} - \frac{299}{5000} a^{11} + \frac{893}{10000} a^{10} + \frac{2301}{10000} a^{9} - \frac{134}{625} a^{8} + \frac{28}{125} a^{7} - \frac{899}{2000} a^{6} + \frac{59}{400} a^{5} + \frac{71}{200} a^{4} + \frac{3}{8} a^{3} - \frac{13}{40} a^{2} + \frac{5}{16} a - \frac{1}{4}$, $\frac{1}{189638690000} a^{19} + \frac{7664703}{189638690000} a^{18} + \frac{18357793}{189638690000} a^{17} + \frac{25944121}{94819345000} a^{16} - \frac{8658097}{5125370000} a^{15} - \frac{7166561}{23704836250} a^{14} - \frac{335243159}{47409672500} a^{13} + \frac{2764472659}{189638690000} a^{12} - \frac{1450604167}{37927738000} a^{11} - \frac{2938193546}{11852418125} a^{10} - \frac{845880933}{11852418125} a^{9} - \frac{21882458989}{189638690000} a^{8} + \frac{16675446581}{37927738000} a^{7} - \frac{1147589186}{2370483625} a^{6} - \frac{219403989}{1896386900} a^{5} - \frac{2256520847}{7585547600} a^{4} - \frac{219733401}{758554760} a^{3} + \frac{575777933}{1517109520} a^{2} + \frac{59842223}{303421904} a - \frac{23311225}{303421904}$

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:  $15$
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:  \( 1609341389.48 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1013:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3686400
The 114 conjugacy class representatives for t20n1013 are not computed
Character table for t20n1013 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.109268775200000.1

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 32 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$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}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$29$29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.5.0.1$x^{5} - x + 11$$1$$5$$0$$C_5$$[\ ]^{5}$
29.5.0.1$x^{5} - x + 11$$1$$5$$0$$C_5$$[\ ]^{5}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.6.4.3$x^{6} + 713 x^{3} + 138384$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$