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Properties

Label 20.12.6684829064...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{59}\cdot 3^{16}\cdot 5^{20}\cdot 7^{10}$
Root discriminant $246.18$
Ramified primes $2, 3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T654

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-82448, 2648640, -17382400, -10771040, 91239560, -52038544, -16284640, 6431480, 1432545, 2794600, -605440, -271700, -94900, 20280, 9750, 1008, 170, -180, -20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 20*x^18 - 180*x^17 + 170*x^16 + 1008*x^15 + 9750*x^14 + 20280*x^13 - 94900*x^12 - 271700*x^11 - 605440*x^10 + 2794600*x^9 + 1432545*x^8 + 6431480*x^7 - 16284640*x^6 - 52038544*x^5 + 91239560*x^4 - 10771040*x^3 - 17382400*x^2 + 2648640*x - 82448)
 
gp: K = bnfinit(x^20 - 20*x^18 - 180*x^17 + 170*x^16 + 1008*x^15 + 9750*x^14 + 20280*x^13 - 94900*x^12 - 271700*x^11 - 605440*x^10 + 2794600*x^9 + 1432545*x^8 + 6431480*x^7 - 16284640*x^6 - 52038544*x^5 + 91239560*x^4 - 10771040*x^3 - 17382400*x^2 + 2648640*x - 82448, 1)
 

Normalized defining polynomial

\( x^{20} - 20 x^{18} - 180 x^{17} + 170 x^{16} + 1008 x^{15} + 9750 x^{14} + 20280 x^{13} - 94900 x^{12} - 271700 x^{11} - 605440 x^{10} + 2794600 x^{9} + 1432545 x^{8} + 6431480 x^{7} - 16284640 x^{6} - 52038544 x^{5} + 91239560 x^{4} - 10771040 x^{3} - 17382400 x^{2} + 2648640 x - 82448 \)

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:  \(668482906464715218862945075200000000000000000000=2^{59}\cdot 3^{16}\cdot 5^{20}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $246.18$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{14} a^{14} + \frac{1}{7} a^{12} + \frac{1}{7} a^{11} + \frac{3}{7} a^{10} + \frac{1}{7} a^{9} - \frac{3}{7} a^{8} - \frac{2}{7} a^{6} - \frac{2}{7} a^{4} + \frac{1}{7} a^{3} + \frac{3}{14} a^{2} - \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{14} a^{15} + \frac{1}{7} a^{13} + \frac{1}{7} a^{12} + \frac{3}{7} a^{11} + \frac{1}{7} a^{10} - \frac{3}{7} a^{9} - \frac{2}{7} a^{7} - \frac{2}{7} a^{5} + \frac{1}{7} a^{4} + \frac{3}{14} a^{3} - \frac{3}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{28} a^{16} - \frac{3}{7} a^{13} + \frac{1}{14} a^{12} + \frac{3}{7} a^{11} + \frac{5}{14} a^{10} - \frac{1}{7} a^{9} + \frac{2}{7} a^{8} + \frac{1}{7} a^{6} - \frac{3}{7} a^{5} - \frac{3}{28} a^{4} + \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{56} a^{17} - \frac{13}{28} a^{13} + \frac{1}{7} a^{12} + \frac{3}{28} a^{11} - \frac{2}{7} a^{10} + \frac{1}{14} a^{9} + \frac{3}{14} a^{8} - \frac{3}{7} a^{7} + \frac{3}{7} a^{6} + \frac{25}{56} a^{5} - \frac{2}{7} a^{4} - \frac{3}{14} a^{3} + \frac{5}{14} a^{2} - \frac{3}{14} a - \frac{3}{7}$, $\frac{1}{392} a^{18} + \frac{3}{196} a^{16} - \frac{3}{98} a^{15} - \frac{5}{196} a^{14} + \frac{24}{49} a^{13} - \frac{71}{196} a^{12} + \frac{2}{49} a^{11} + \frac{24}{49} a^{10} + \frac{37}{98} a^{9} - \frac{9}{49} a^{8} - \frac{19}{49} a^{7} + \frac{153}{392} a^{6} - \frac{19}{49} a^{5} - \frac{31}{196} a^{4} - \frac{16}{49} a^{3} + \frac{29}{98} a^{2} - \frac{10}{49} a + \frac{13}{49}$, $\frac{1}{2019159729203228537451729992123075326226668146032753804009008843624304} a^{19} - \frac{76496377318642240132085931519754414772487491664268103029553351893}{504789932300807134362932498030768831556667036508188451002252210906076} a^{18} - \frac{3281380042970585254132965263442156882111388436989109306526950406183}{1009579864601614268725864996061537663113334073016376902004504421812152} a^{17} - \frac{8160001774316368460467592817321934270443287852999809948057735002429}{504789932300807134362932498030768831556667036508188451002252210906076} a^{16} - \frac{7123354143836519430383581194340526818285928422928110807978935286117}{1009579864601614268725864996061537663113334073016376902004504421812152} a^{15} - \frac{385348463116561981455371031047111929782433607009877210539374836207}{18028211867885969084390446358241743984166679875292444678651864675217} a^{14} - \frac{245900003497994013660370911048295706825242674223130729962920521817515}{1009579864601614268725864996061537663113334073016376902004504421812152} a^{13} - \frac{5204435208993528300259997322510978043182137691963486688821399915505}{18028211867885969084390446358241743984166679875292444678651864675217} a^{12} + \frac{598269006452735881092012784027667886003678233162722794039998585997}{252394966150403567181466249015384415778333518254094225501126105453038} a^{11} - \frac{207843293329975028866727424145878783559851643979505826549844077554953}{504789932300807134362932498030768831556667036508188451002252210906076} a^{10} - \frac{19716265284845147681283799973727336813327524770104265794551878808849}{126197483075201783590733124507692207889166759127047112750563052726519} a^{9} - \frac{1182556963622435614102229658215477426836529902681115705188948132223}{2575458838269424154912920908320249140595239982184634954093123525031} a^{8} - \frac{408073606264464250013645532382112303935373467838206892043759433253679}{2019159729203228537451729992123075326226668146032753804009008843624304} a^{7} + \frac{41176064530523618228904389659272869039952684015172515234344755825061}{504789932300807134362932498030768831556667036508188451002252210906076} a^{6} + \frac{127230429986999495429583911495299397701419602357613594309636107138803}{1009579864601614268725864996061537663113334073016376902004504421812152} a^{5} - \frac{2891724354198807267072631528440673160518601466705099957423692136763}{126197483075201783590733124507692207889166759127047112750563052726519} a^{4} + \frac{252122142116406041251895800247312846416112945515012084410299730851079}{504789932300807134362932498030768831556667036508188451002252210906076} a^{3} + \frac{19841023376812237426194613030535608349922443329257870214128350048448}{126197483075201783590733124507692207889166759127047112750563052726519} a^{2} + \frac{111497051056917504325773582137263851967674995900964366837986047336605}{252394966150403567181466249015384415778333518254094225501126105453038} a - \frac{106892670272857396196175251973656220825466664734921386689846673410}{126197483075201783590733124507692207889166759127047112750563052726519}$

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:  \( 240115094787000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T654:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 70 conjugacy class representatives for t20n654 are not computed
Character table for t20n654 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 10.6.564586122240000000000.1

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 24 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 R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ $20$

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$2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.8.26.4$x^{8} + 8 x^{7} + 12 x^{6} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 2$$8$$1$$26$$C_2^2:C_4$$[2, 3, 7/2, 4]$
2.8.22.2$x^{8} + 10 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.12.12.12$x^{12} + 165 x^{10} - 312 x^{9} - 288 x^{8} - 180 x^{7} - 36 x^{6} - 135 x^{5} - 243 x^{4} + 54 x^{3} + 81 x^{2} + 81 x - 162$$3$$4$$12$12T41$[3/2, 3/2]_{2}^{4}$
5Data not computed
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.6.0.1$x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
7.10.8.1$x^{10} - 7 x^{5} + 147$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$