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Properties

Label 20.20.8308610292...3888.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{30}\cdot 3^{10}\cdot 7^{8}\cdot 11^{8}\cdot 13^{9}$
Root discriminant $88.30$
Ramified primes $2, 3, 7, 11, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{10}^2:C_2^2$ (as 20T106)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6157, 15576, -248460, -643258, 1575260, 1939996, -3070474, -1981016, 2376321, 972558, -929382, -258790, 204088, 38676, -26030, -3176, 1884, 130, -70, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 70*x^18 + 130*x^17 + 1884*x^16 - 3176*x^15 - 26030*x^14 + 38676*x^13 + 204088*x^12 - 258790*x^11 - 929382*x^10 + 972558*x^9 + 2376321*x^8 - 1981016*x^7 - 3070474*x^6 + 1939996*x^5 + 1575260*x^4 - 643258*x^3 - 248460*x^2 + 15576*x + 6157)
 
gp: K = bnfinit(x^20 - 2*x^19 - 70*x^18 + 130*x^17 + 1884*x^16 - 3176*x^15 - 26030*x^14 + 38676*x^13 + 204088*x^12 - 258790*x^11 - 929382*x^10 + 972558*x^9 + 2376321*x^8 - 1981016*x^7 - 3070474*x^6 + 1939996*x^5 + 1575260*x^4 - 643258*x^3 - 248460*x^2 + 15576*x + 6157, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 70 x^{18} + 130 x^{17} + 1884 x^{16} - 3176 x^{15} - 26030 x^{14} + 38676 x^{13} + 204088 x^{12} - 258790 x^{11} - 929382 x^{10} + 972558 x^{9} + 2376321 x^{8} - 1981016 x^{7} - 3070474 x^{6} + 1939996 x^{5} + 1575260 x^{4} - 643258 x^{3} - 248460 x^{2} + 15576 x + 6157 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(830861029216363101082988400989215653888=2^{30}\cdot 3^{10}\cdot 7^{8}\cdot 11^{8}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.30$
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:  $2, 3, 7, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{13} a^{17} + \frac{5}{13} a^{16} - \frac{4}{13} a^{15} - \frac{6}{13} a^{13} - \frac{6}{13} a^{12} + \frac{2}{13} a^{11} + \frac{6}{13} a^{10} - \frac{4}{13} a^{9} - \frac{4}{13} a^{8} - \frac{3}{13} a^{7} + \frac{1}{13} a^{6} + \frac{5}{13} a^{5} - \frac{6}{13} a^{4} - \frac{5}{13} a^{3} - \frac{5}{13} a^{2} - \frac{2}{13} a + \frac{6}{13}$, $\frac{1}{13} a^{18} - \frac{3}{13} a^{16} - \frac{6}{13} a^{15} - \frac{6}{13} a^{14} - \frac{2}{13} a^{13} + \frac{6}{13} a^{12} - \frac{4}{13} a^{11} + \frac{5}{13} a^{10} + \frac{3}{13} a^{9} + \frac{4}{13} a^{8} + \frac{3}{13} a^{7} - \frac{5}{13} a^{5} - \frac{1}{13} a^{4} - \frac{6}{13} a^{3} - \frac{3}{13} a^{2} + \frac{3}{13} a - \frac{4}{13}$, $\frac{1}{904043278151745018266694781643477704124895646267} a^{19} - \frac{5195056447633498700426827329107821787791612867}{904043278151745018266694781643477704124895646267} a^{18} - \frac{782168831860845836435338594417462981453510903}{39306229484858479055073686158412074092386767229} a^{17} + \frac{196325035090802268566871175112559550277839528380}{904043278151745018266694781643477704124895646267} a^{16} + \frac{360049733146611993602778337295700492697357196426}{904043278151745018266694781643477704124895646267} a^{15} - \frac{402319290057105477311805203946795278384274372363}{904043278151745018266694781643477704124895646267} a^{14} + \frac{14236282008705421804993120420349607504050285879}{69541790627057309097438060126421361855761203559} a^{13} + \frac{296169805111071347065485842848357020134304855869}{904043278151745018266694781643477704124895646267} a^{12} + \frac{289498414003180111090698196224067162654891343783}{904043278151745018266694781643477704124895646267} a^{11} + \frac{67925442839010687319918670264623982435618523448}{904043278151745018266694781643477704124895646267} a^{10} - \frac{32790469232295184529964515558576047572139373872}{69541790627057309097438060126421361855761203559} a^{9} - \frac{436377574663661994719468278753103678650023827901}{904043278151745018266694781643477704124895646267} a^{8} - \frac{68639879170688213164413340272877534553699613882}{904043278151745018266694781643477704124895646267} a^{7} + \frac{5483629854560795968782790307528401441863985455}{904043278151745018266694781643477704124895646267} a^{6} + \frac{29532784423189839996052995943787231401425640229}{904043278151745018266694781643477704124895646267} a^{5} - \frac{69132323615808412901887244203433227348739623621}{904043278151745018266694781643477704124895646267} a^{4} + \frac{152819959170873390087583373905666905259857824713}{904043278151745018266694781643477704124895646267} a^{3} - \frac{86676868633113790247647969268439980363793544187}{904043278151745018266694781643477704124895646267} a^{2} - \frac{154132755467317830314328373862526712820440354006}{904043278151745018266694781643477704124895646267} a + \frac{21120922655675361457341410563639154100434553}{146831781411685076866443849544173737879632231}$

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:  $19$
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:  \( 16168836073800 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{10}^2:C_2^2$ (as 20T106):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 46 conjugacy class representatives for $C_{10}^2:C_2^2$
Character table for $C_{10}^2:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.7488.1, 10.10.249828821987576832.1

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 R $20$ R R R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
3Data not computed
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.10.8.2$x^{10} + 143 x^{5} + 5929$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{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}$