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Properties

Label 20.0.75045768722...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{26}\cdot 31^{8}$
Root discriminant $55.43$
Ramified primes $3, 5, 31$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $D_5^2$ (as 20T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![55238851, 70381245, -32489355, -43775185, 15953485, 7848131, 448205, 1129650, -1315060, 111430, 59367, -105110, 56490, -6250, -660, 1594, -460, 30, 5, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 5*x^18 + 30*x^17 - 460*x^16 + 1594*x^15 - 660*x^14 - 6250*x^13 + 56490*x^12 - 105110*x^11 + 59367*x^10 + 111430*x^9 - 1315060*x^8 + 1129650*x^7 + 448205*x^6 + 7848131*x^5 + 15953485*x^4 - 43775185*x^3 - 32489355*x^2 + 70381245*x + 55238851)
 
gp: K = bnfinit(x^20 - 5*x^19 + 5*x^18 + 30*x^17 - 460*x^16 + 1594*x^15 - 660*x^14 - 6250*x^13 + 56490*x^12 - 105110*x^11 + 59367*x^10 + 111430*x^9 - 1315060*x^8 + 1129650*x^7 + 448205*x^6 + 7848131*x^5 + 15953485*x^4 - 43775185*x^3 - 32489355*x^2 + 70381245*x + 55238851, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 5 x^{18} + 30 x^{17} - 460 x^{16} + 1594 x^{15} - 660 x^{14} - 6250 x^{13} + 56490 x^{12} - 105110 x^{11} + 59367 x^{10} + 111430 x^{9} - 1315060 x^{8} + 1129650 x^{7} + 448205 x^{6} + 7848131 x^{5} + 15953485 x^{4} - 43775185 x^{3} - 32489355 x^{2} + 70381245 x + 55238851 \)

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:  \(75045768722673667967319488525390625=3^{10}\cdot 5^{26}\cdot 31^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.43$
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:  $3, 5, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{1}{7} a^{11} + \frac{3}{7} a^{10} - \frac{3}{7} a^{9} + \frac{2}{7} a^{8} + \frac{1}{7} a^{6} - \frac{3}{7} a^{5} - \frac{1}{7} a^{4} + \frac{2}{7} a^{3} - \frac{1}{7} a^{2} + \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{13} + \frac{2}{7} a^{11} - \frac{1}{7} a^{9} + \frac{2}{7} a^{8} + \frac{1}{7} a^{7} - \frac{2}{7} a^{6} + \frac{3}{7} a^{5} + \frac{1}{7} a^{4} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{21} a^{14} + \frac{1}{21} a^{13} - \frac{10}{21} a^{11} - \frac{1}{3} a^{10} - \frac{8}{21} a^{8} - \frac{8}{21} a^{7} - \frac{8}{21} a^{6} + \frac{1}{7} a^{5} - \frac{10}{21} a^{4} - \frac{1}{21} a^{3} + \frac{3}{7} a^{2} + \frac{1}{21} a - \frac{2}{21}$, $\frac{1}{21} a^{15} - \frac{1}{21} a^{13} - \frac{1}{21} a^{12} - \frac{2}{7} a^{11} - \frac{8}{21} a^{10} + \frac{1}{3} a^{9} - \frac{1}{7} a^{8} - \frac{1}{21} a^{6} + \frac{2}{21} a^{5} + \frac{1}{3} a^{3} + \frac{4}{21} a^{2} + \frac{1}{7} a - \frac{1}{21}$, $\frac{1}{147} a^{16} - \frac{2}{147} a^{15} - \frac{2}{147} a^{14} - \frac{1}{49} a^{13} - \frac{1}{21} a^{12} - \frac{31}{147} a^{11} - \frac{2}{7} a^{10} - \frac{5}{147} a^{9} + \frac{23}{147} a^{8} + \frac{67}{147} a^{7} - \frac{2}{49} a^{6} + \frac{5}{21} a^{5} + \frac{38}{147} a^{4} - \frac{13}{49} a^{3} + \frac{25}{147} a^{2} - \frac{53}{147} a + \frac{55}{147}$, $\frac{1}{147} a^{17} + \frac{1}{147} a^{15} + \frac{8}{147} a^{13} - \frac{10}{147} a^{12} - \frac{23}{49} a^{11} - \frac{68}{147} a^{10} + \frac{62}{147} a^{9} + \frac{5}{49} a^{8} - \frac{18}{49} a^{7} - \frac{40}{147} a^{6} - \frac{67}{147} a^{5} - \frac{18}{49} a^{4} - \frac{53}{147} a^{3} - \frac{59}{147} a^{2} + \frac{61}{147} a + \frac{68}{147}$, $\frac{1}{20839804409151} a^{18} + \frac{659383222}{330790546177} a^{17} + \frac{21749369104}{6946601469717} a^{16} + \frac{86574279254}{6946601469717} a^{15} + \frac{443739573470}{20839804409151} a^{14} - \frac{1296244423021}{20839804409151} a^{13} - \frac{86631466504}{6946601469717} a^{12} + \frac{4611258228013}{20839804409151} a^{11} + \frac{9259984482440}{20839804409151} a^{10} - \frac{1298839885738}{6946601469717} a^{9} - \frac{410893484644}{1603061877627} a^{8} + \frac{1518083742374}{20839804409151} a^{7} - \frac{2251411091155}{20839804409151} a^{6} - \frac{638130180918}{2315533823239} a^{5} - \frac{10361577959318}{20839804409151} a^{4} - \frac{70688592965}{20839804409151} a^{3} - \frac{6780412654415}{20839804409151} a^{2} - \frac{6979638660403}{20839804409151} a + \frac{3840975562373}{20839804409151}$, $\frac{1}{17009327048138634680972647740960741561256910242956036697941} a^{19} - \frac{55113540861954724705201368028366891532356229}{5669775682712878226990882580320247187085636747652012232647} a^{18} - \frac{5029263264222853136126451781157550238005102289807118174}{1889925227570959408996960860106749062361878915884004077549} a^{17} + \frac{136297184736858670624286318496008162484112570723373110}{436136590977913709768529429255403629775818211357847094819} a^{16} + \frac{17143959947177468451298164768815456258163934754637030320}{1308409772933741129305588287766210889327454634073541284457} a^{15} + \frac{30244242340529892867011030306033958799232565625989047}{186915681847677304186512612538030127046779233439077326351} a^{14} + \frac{2991356529620288965873939771009623391903256870323416766}{809967954673268318141554654331463883869376678236001747521} a^{13} - \frac{56903728504900910347245021943984076783063620019866525322}{2429903864019804954424663962994391651608130034708005242563} a^{12} + \frac{4403117128929724618278189149385608984072797714509192151565}{17009327048138634680972647740960741561256910242956036697941} a^{11} + \frac{1866961192530448933581396218018015429712724471410304761661}{5669775682712878226990882580320247187085636747652012232647} a^{10} + \frac{1129058511942944219158910790549859670882630277305242782419}{17009327048138634680972647740960741561256910242956036697941} a^{9} - \frac{5973352313818365177466200477085850296249242819932258998540}{17009327048138634680972647740960741561256910242956036697941} a^{8} + \frac{3056894745951779203565750648988777374825520176861404933700}{17009327048138634680972647740960741561256910242956036697941} a^{7} - \frac{1340516440401984617573265922010229095229027822499501087567}{5669775682712878226990882580320247187085636747652012232647} a^{6} + \frac{8221793999396169045175001061920212882476211083018295685074}{17009327048138634680972647740960741561256910242956036697941} a^{5} - \frac{29367310292511562304410061886226507328391118839893905184}{17009327048138634680972647740960741561256910242956036697941} a^{4} + \frac{7007887163685178342152307915014435653174438295172921772614}{17009327048138634680972647740960741561256910242956036697941} a^{3} - \frac{11968888367067473768075814643246026819506380266975666728}{34223998084785985273586816380202699318424366686028242853} a^{2} + \frac{1170789670860907464059884538278638774467763721752919406519}{2429903864019804954424663962994391651608130034708005242563} a + \frac{911076723941822476289726462356953037348489473500490316933}{5669775682712878226990882580320247187085636747652012232647}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{365258343734433380611056131441581655}{4999774636833680677039311606729464615097537} a^{19} + \frac{712047643339558004792473396119850055}{1666591545611226892346437202243154871699179} a^{18} - \frac{1281413633132608486730366014714905650}{1666591545611226892346437202243154871699179} a^{17} - \frac{747608263025333709156270329790663580}{555530515203742297448812400747718290566393} a^{16} + \frac{172663743815190190959380176923691697339}{4999774636833680677039311606729464615097537} a^{15} - \frac{733354908380152741141780866342400845055}{4999774636833680677039311606729464615097537} a^{14} + \frac{317276678333288966421967994530033606315}{1666591545611226892346437202243154871699179} a^{13} + \frac{1170828767021066171847377146098030854845}{4999774636833680677039311606729464615097537} a^{12} - \frac{3059754271162097181383794185052988749645}{714253519547668668148473086675637802156791} a^{11} + \frac{2724649732245482602759545520741593801617}{238084506515889556049491028891879267385597} a^{10} - \frac{80244458150637625684414718665717375701820}{4999774636833680677039311606729464615097537} a^{9} + \frac{45771859249144718127136517934808034813150}{4999774636833680677039311606729464615097537} a^{8} + \frac{414057368913300936049660361090776971598160}{4999774636833680677039311606729464615097537} a^{7} - \frac{246447008195094229290939606555893587171645}{1666591545611226892346437202243154871699179} a^{6} + \frac{596946143769814560407387216064008044873078}{4999774636833680677039311606729464615097537} a^{5} - \frac{3415373291894769166531207041353498704940165}{4999774636833680677039311606729464615097537} a^{4} - \frac{60272381008873502409405439885707513777665}{102036217078238381164067583810805400308113} a^{3} + \frac{229863512768314187258456296640082846929110}{70419361082164516578018473334217811480247} a^{2} - \frac{4447289813486818492641475773988419773328055}{4999774636833680677039311606729464615097537} a - \frac{1704574943378131688421234162515889451527461}{555530515203742297448812400747718290566393} \) (order $6$)
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:  \( 1625055692.34 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5^2$ (as 20T28):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 16 conjugacy class representatives for $D_5^2$
Character table for $D_5^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 10.0.54788965576171875.1 x5

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 sibling: data not computed
Degree 25 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5Data not computed
$31$31.5.4.2$x^{5} + 217$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.0.1$x^{5} - x + 10$$1$$5$$0$$C_5$$[\ ]^{5}$
31.5.0.1$x^{5} - x + 10$$1$$5$$0$$C_5$$[\ ]^{5}$
31.5.4.2$x^{5} + 217$$5$$1$$4$$C_5$$[\ ]_{5}$