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Properties

Label 20.0.32758971529...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{10}\cdot 13^{15}$
Root discriminant $26.65$
Ramified primes $2, 5, 13$
Class number $2$
Class group $[2]$
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 351, 1980, 3726, 5774, 3161, 8626, 14, 3461, 362, 664, 538, 568, 272, 102, -58, 41, -7, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 12*x^18 - 7*x^17 + 41*x^16 - 58*x^15 + 102*x^14 + 272*x^13 + 568*x^12 + 538*x^11 + 664*x^10 + 362*x^9 + 3461*x^8 + 14*x^7 + 8626*x^6 + 3161*x^5 + 5774*x^4 + 3726*x^3 + 1980*x^2 + 351*x + 81)
 
gp: K = bnfinit(x^20 + 12*x^18 - 7*x^17 + 41*x^16 - 58*x^15 + 102*x^14 + 272*x^13 + 568*x^12 + 538*x^11 + 664*x^10 + 362*x^9 + 3461*x^8 + 14*x^7 + 8626*x^6 + 3161*x^5 + 5774*x^4 + 3726*x^3 + 1980*x^2 + 351*x + 81, 1)
 

Normalized defining polynomial

\( x^{20} + 12 x^{18} - 7 x^{17} + 41 x^{16} - 58 x^{15} + 102 x^{14} + 272 x^{13} + 568 x^{12} + 538 x^{11} + 664 x^{10} + 362 x^{9} + 3461 x^{8} + 14 x^{7} + 8626 x^{6} + 3161 x^{5} + 5774 x^{4} + 3726 x^{3} + 1980 x^{2} + 351 x + 81 \)

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:  \(32758971529018084480000000000=2^{16}\cdot 5^{10}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.65$
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:  $2, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{1}{3} a^{6} + \frac{4}{9} a^{5} - \frac{1}{3} a^{4} - \frac{1}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{13} - \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{1}{3} a^{7} + \frac{4}{9} a^{6} - \frac{1}{3} a^{5} - \frac{1}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{117} a^{16} + \frac{2}{117} a^{15} - \frac{5}{117} a^{14} + \frac{1}{9} a^{13} - \frac{2}{13} a^{12} - \frac{4}{117} a^{11} - \frac{1}{39} a^{10} + \frac{1}{39} a^{9} + \frac{17}{117} a^{8} + \frac{43}{117} a^{7} + \frac{38}{117} a^{6} + \frac{17}{117} a^{5} + \frac{8}{39} a^{4} - \frac{2}{9} a^{3} + \frac{7}{39} a^{2} - \frac{5}{13} a + \frac{1}{13}$, $\frac{1}{1053} a^{17} - \frac{1}{351} a^{16} + \frac{50}{1053} a^{15} - \frac{40}{1053} a^{14} - \frac{5}{39} a^{13} + \frac{11}{117} a^{12} + \frac{23}{351} a^{11} - \frac{34}{1053} a^{10} + \frac{106}{1053} a^{9} - \frac{55}{1053} a^{8} + \frac{2}{117} a^{7} + \frac{29}{351} a^{6} - \frac{451}{1053} a^{5} - \frac{484}{1053} a^{4} - \frac{265}{1053} a^{3} + \frac{145}{351} a^{2} + \frac{1}{3} a + \frac{7}{39}$, $\frac{1}{752895} a^{18} + \frac{113}{250965} a^{17} + \frac{1004}{752895} a^{16} + \frac{4523}{752895} a^{15} - \frac{29}{27885} a^{14} + \frac{4109}{27885} a^{13} - \frac{20188}{250965} a^{12} - \frac{59236}{752895} a^{11} - \frac{7798}{150579} a^{10} - \frac{84808}{752895} a^{9} - \frac{3709}{27885} a^{8} + \frac{3239}{250965} a^{7} + \frac{302156}{752895} a^{6} - \frac{17591}{150579} a^{5} - \frac{325363}{752895} a^{4} + \frac{111973}{250965} a^{3} + \frac{15248}{83655} a^{2} + \frac{2473}{9295} a - \frac{4416}{9295}$, $\frac{1}{322558356819009641660806905} a^{19} + \frac{34706628957180382063}{107519452273003213886935635} a^{18} - \frac{13448376908908041597754}{35839817424334404628978545} a^{17} - \frac{82960165333193469124547}{29323486983546331060073355} a^{16} + \frac{232004166127647499950467}{322558356819009641660806905} a^{15} - \frac{6868422134140035548143447}{322558356819009641660806905} a^{14} - \frac{11294557544795789664344533}{107519452273003213886935635} a^{13} + \frac{3508784857172994362626844}{322558356819009641660806905} a^{12} - \frac{307882943655282398643517}{4962436258753994487089337} a^{11} + \frac{13951764084244462649948902}{322558356819009641660806905} a^{10} + \frac{3169165771776026551436662}{322558356819009641660806905} a^{9} - \frac{20716670476978857647634328}{322558356819009641660806905} a^{8} + \frac{116918760264818426123885531}{322558356819009641660806905} a^{7} - \frac{12075783469745754271554905}{64511671363801928332161381} a^{6} + \frac{42793831568265371491385002}{322558356819009641660806905} a^{5} + \frac{24664595709917985227144879}{322558356819009641660806905} a^{4} - \frac{8636180981461191148684948}{322558356819009641660806905} a^{3} - \frac{4149915107470032909951869}{107519452273003213886935635} a^{2} + \frac{168726883913641944518954}{362018357821559642716955} a - \frac{721592734085381818054327}{2389321161622293641931903}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1421636.55011 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.54925.1, 5.1.878800.1 x5, 10.2.10039762720000.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.878800.1
Degree 10 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 R ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$13$13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$