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Properties

Label 20.0.56136275023...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 3^{16}\cdot 5^{37}\cdot 29^{5}\cdot 71^{10}$
Root discriminant $2440.41$
Ramified primes $2, 3, 5, 29, 71$
Class number Not computed
Class group Not computed
Galois group $D_4\times F_5$ (as 20T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4004337882200625, 1983734041725000, 690460909256250, 235380065535000, 15622931548125, -21144057010200, 2300599989000, -1145774781000, 58085116650, 9144603000, 1809909476, 430677000, 45780530, 351000, 854360, -5400, 13285, 0, 130, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 130*x^18 + 13285*x^16 - 5400*x^15 + 854360*x^14 + 351000*x^13 + 45780530*x^12 + 430677000*x^11 + 1809909476*x^10 + 9144603000*x^9 + 58085116650*x^8 - 1145774781000*x^7 + 2300599989000*x^6 - 21144057010200*x^5 + 15622931548125*x^4 + 235380065535000*x^3 + 690460909256250*x^2 + 1983734041725000*x + 4004337882200625)
 
gp: K = bnfinit(x^20 + 130*x^18 + 13285*x^16 - 5400*x^15 + 854360*x^14 + 351000*x^13 + 45780530*x^12 + 430677000*x^11 + 1809909476*x^10 + 9144603000*x^9 + 58085116650*x^8 - 1145774781000*x^7 + 2300599989000*x^6 - 21144057010200*x^5 + 15622931548125*x^4 + 235380065535000*x^3 + 690460909256250*x^2 + 1983734041725000*x + 4004337882200625, 1)
 

Normalized defining polynomial

\( x^{20} + 130 x^{18} + 13285 x^{16} - 5400 x^{15} + 854360 x^{14} + 351000 x^{13} + 45780530 x^{12} + 430677000 x^{11} + 1809909476 x^{10} + 9144603000 x^{9} + 58085116650 x^{8} - 1145774781000 x^{7} + 2300599989000 x^{6} - 21144057010200 x^{5} + 15622931548125 x^{4} + 235380065535000 x^{3} + 690460909256250 x^{2} + 1983734041725000 x + 4004337882200625 \)

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:  \(56136275023201124305517265191938066406250000000000000000000000000000=2^{28}\cdot 3^{16}\cdot 5^{37}\cdot 29^{5}\cdot 71^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2440.41$
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:  $2, 3, 5, 29, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{6} + \frac{1}{6} a^{4} + \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} + \frac{1}{6} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{8} + \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{60} a^{9} - \frac{1}{12} a^{8} - \frac{1}{30} a^{7} - \frac{1}{30} a^{5} - \frac{1}{2} a^{4} + \frac{1}{10} a^{3} + \frac{1}{3} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{240} a^{10} + \frac{13}{240} a^{8} + \frac{3}{40} a^{6} - \frac{1}{4} a^{5} - \frac{47}{120} a^{4} - \frac{1}{2} a^{3} - \frac{23}{48} a^{2} - \frac{1}{4} a - \frac{5}{16}$, $\frac{1}{240} a^{11} + \frac{1}{240} a^{9} - \frac{1}{12} a^{8} + \frac{1}{120} a^{7} - \frac{1}{12} a^{6} + \frac{1}{24} a^{5} - \frac{1}{3} a^{4} + \frac{13}{240} a^{3} + \frac{1}{4} a^{2} + \frac{7}{16} a + \frac{1}{4}$, $\frac{1}{1440} a^{12} + \frac{1}{720} a^{10} + \frac{5}{96} a^{8} - \frac{1}{24} a^{7} - \frac{13}{360} a^{6} + \frac{5}{24} a^{5} + \frac{199}{1440} a^{4} + \frac{5}{24} a^{3} + \frac{7}{16} a^{2} - \frac{3}{8} a + \frac{5}{32}$, $\frac{1}{1440} a^{13} + \frac{1}{720} a^{11} + \frac{1}{480} a^{9} + \frac{1}{24} a^{8} + \frac{23}{360} a^{7} + \frac{1}{24} a^{6} + \frac{343}{1440} a^{5} - \frac{11}{24} a^{4} + \frac{11}{80} a^{3} + \frac{1}{8} a^{2} - \frac{3}{32} a + \frac{1}{4}$, $\frac{1}{1440} a^{14} - \frac{1}{1440} a^{10} - \frac{1}{120} a^{9} + \frac{31}{720} a^{8} + \frac{7}{120} a^{7} - \frac{11}{480} a^{6} + \frac{7}{120} a^{5} + \frac{1}{36} a^{4} + \frac{29}{120} a^{3} + \frac{35}{96} a^{2} + \frac{1}{4} a + \frac{7}{16}$, $\frac{1}{21600} a^{15} - \frac{1}{4320} a^{13} - \frac{7}{4320} a^{11} - \frac{29}{4320} a^{9} - \frac{1}{24} a^{8} - \frac{233}{4320} a^{7} + \frac{3221}{21600} a^{5} + \frac{1}{4} a^{4} - \frac{29}{480} a^{3} - \frac{1}{3} a^{2} - \frac{13}{32} a - \frac{3}{8}$, $\frac{1}{129600} a^{16} + \frac{1}{3240} a^{14} + \frac{1}{12960} a^{12} - \frac{1}{480} a^{11} - \frac{19}{12960} a^{10} + \frac{1}{160} a^{9} + \frac{23}{12960} a^{8} + \frac{1}{48} a^{7} - \frac{503}{16200} a^{6} - \frac{29}{240} a^{5} + \frac{239}{1440} a^{4} - \frac{209}{480} a^{3} + \frac{11}{32} a^{2} - \frac{41}{96} a - \frac{15}{64}$, $\frac{1}{648000} a^{17} + \frac{1}{64800} a^{15} + \frac{1}{4050} a^{13} + \frac{43}{32400} a^{11} - \frac{1}{480} a^{10} + \frac{229}{32400} a^{9} - \frac{7}{160} a^{8} - \frac{25037}{324000} a^{7} + \frac{19}{240} a^{6} - \frac{761}{5400} a^{5} + \frac{7}{16} a^{4} - \frac{7}{48} a^{3} + \frac{73}{160} a^{2} - \frac{1}{64} a + \frac{9}{32}$, $\frac{1}{165082242787117331754551650906118808024000} a^{18} + \frac{26444858594445796185420046715140477}{55027414262372443918183883635372936008000} a^{17} + \frac{7535142522936328341031830668338373}{2063528034838966646931895636326485100300} a^{16} - \frac{14872283115607249291246180161325829}{5502741426237244391818388363537293600800} a^{15} + \frac{3750333656413902763329130106117126371}{16508224278711733175455165090611880802400} a^{14} - \frac{744563414400583909498061795924154623}{5502741426237244391818388363537293600800} a^{13} - \frac{1061916310854326057546936300986411549}{16508224278711733175455165090611880802400} a^{12} - \frac{5442597973142799161373257467113891643}{5502741426237244391818388363537293600800} a^{11} + \frac{107490351155186071335165317525154571}{2063528034838966646931895636326485100300} a^{10} + \frac{4112990028264691019227523558289195509}{1375685356559311097954597090884323400200} a^{9} + \frac{212548062890887010734297899331416495913}{82541121393558665877275825453059404012000} a^{8} - \frac{1085892080510923296522400342437669649949}{27513707131186221959091941817686468004000} a^{7} + \frac{239542535371604222269726289696979999257}{5502741426237244391818388363537293600800} a^{6} + \frac{69168391526974774387616321917752289219}{366849428415816292787892557569152906720} a^{5} + \frac{125769556529295294533197328310586809}{2717403173450491057688093019030762272} a^{4} - \frac{53544588751391053574876624192223400547}{122283142805272097595964185856384302240} a^{3} + \frac{34232374144256088849039621967753355291}{81522095203514731730642790570922868160} a^{2} - \frac{3574841536644135799422575729813178239}{16304419040702946346128558114184573632} a - \frac{529163099007081610779856686505210217}{2717403173450491057688093019030762272}$, $\frac{1}{9447621794296638556366592293296787584820724479339520997022987425720000} a^{19} - \frac{1602840416813484994120600967}{944762179429663855636659229329678758482072447933952099702298742572000} a^{18} + \frac{900394268015788071708426745522234697467470719168532121432550531}{1889524358859327711273318458659357516964144895867904199404597485144000} a^{17} - \frac{68201481232200425396701875807038026884707242295962679208189881}{37790487177186554225466369173187150339282897917358083988091949702880} a^{16} - \frac{7313650574345475295598076227893717957197669732940325034422564917}{472381089714831927818329614664839379241036223966976049851149371286000} a^{15} + \frac{1062827137420191831746551643610145465411214482734595785453843955}{3779048717718655422546636917318715033928289791735808398809194970288} a^{14} - \frac{6437960289808494418107556466314445334231899808021219264328966277}{29523818107176995488645600916552461202564763997936003115696835705375} a^{13} - \frac{1879314946288864765753612833023739588812749515262084089271968359}{188952435885932771127331845865935751696414489586790419940459748514400} a^{12} + \frac{1353982465923698371493915490607931871733605287468261513555481702803}{944762179429663855636659229329678758482072447933952099702298742572000} a^{11} + \frac{94527891127153829234923271450509680384697892879145600482227571069}{94476217942966385563665922932967875848207244793395209970229874257200} a^{10} + \frac{3538379262551475744530402455600771990971874141261799616352732178113}{4723810897148319278183296146648393792410362239669760498511493712860000} a^{9} - \frac{62808409674829341889736492575059592390879779302492696918643833543767}{944762179429663855636659229329678758482072447933952099702298742572000} a^{8} - \frac{1099631291619059254601296253686424368393269400472661063497395220819}{31492072647655461854555307644322625282735748264465069990076624752400} a^{7} - \frac{454343859321059745000278342829561190336573254022982236637046272183}{31492072647655461854555307644322625282735748264465069990076624752400} a^{6} + \frac{1915305304592871254082777243034876129093356171666338048637021029}{29159326525606909124588247818817245632162729874504694435256134030} a^{5} - \frac{905153621180528193287819809552120339001619801574028276937260207411}{6998238366145658189901179476516138951719055169881126664461472167200} a^{4} + \frac{496664685412084678296164493693971261284812631425174630481274411383}{2799295346458263275960471790606455580687622067952450665784588866880} a^{3} + \frac{26936531840190475130657710329571792316572173196313338642843380545}{93309844881942109198682393020215186022920735598415022192819628896} a^{2} - \frac{91820352727440283743054138987054072021732026406985403679497406593}{186619689763884218397364786040430372045841471196830044385639257792} a + \frac{2261054123366589409344635498254789655396646714351048200883960967}{7775820406828509099890199418351265501910061299867918516068302408}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times F_5$ (as 20T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 25 conjugacy class representatives for $D_4\times F_5$
Character table for $D_4\times F_5$ is not computed

Intermediate fields

\(\Q(\sqrt{-71}) \), 4.0.11695120.4, 5.1.2531250000.5, 10.0.11560106222569335937500000000.1

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ R $20$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$5$5.5.9.2$x^{5} + 55$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.9.2$x^{5} + 55$$5$$1$$9$$F_5$$[9/4]_{4}$
5.10.19.5$x^{10} + 55$$10$$1$$19$$F_5$$[9/4]_{4}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$71$71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$