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Properties

Label 20.0.93480162674...0000.7
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{34}\cdot 11^{10}$
Root discriminant $177.23$
Ramified primes $2, 3, 5, 11$
Class number $15439376$ (GRH)
Class group $[2, 7719688]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![315569310049, 11447764160, 122436468035, 3610501410, 23347784530, 775988298, 2830853065, 119368900, 241238600, 12037680, 15113210, 804900, 737860, 26460, 26360, 236, 910, -20, 5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 5*x^18 - 20*x^17 + 910*x^16 + 236*x^15 + 26360*x^14 + 26460*x^13 + 737860*x^12 + 804900*x^11 + 15113210*x^10 + 12037680*x^9 + 241238600*x^8 + 119368900*x^7 + 2830853065*x^6 + 775988298*x^5 + 23347784530*x^4 + 3610501410*x^3 + 122436468035*x^2 + 11447764160*x + 315569310049)
 
gp: K = bnfinit(x^20 + 5*x^18 - 20*x^17 + 910*x^16 + 236*x^15 + 26360*x^14 + 26460*x^13 + 737860*x^12 + 804900*x^11 + 15113210*x^10 + 12037680*x^9 + 241238600*x^8 + 119368900*x^7 + 2830853065*x^6 + 775988298*x^5 + 23347784530*x^4 + 3610501410*x^3 + 122436468035*x^2 + 11447764160*x + 315569310049, 1)
 

Normalized defining polynomial

\( x^{20} + 5 x^{18} - 20 x^{17} + 910 x^{16} + 236 x^{15} + 26360 x^{14} + 26460 x^{13} + 737860 x^{12} + 804900 x^{11} + 15113210 x^{10} + 12037680 x^{9} + 241238600 x^{8} + 119368900 x^{7} + 2830853065 x^{6} + 775988298 x^{5} + 23347784530 x^{4} + 3610501410 x^{3} + 122436468035 x^{2} + 11447764160 x + 315569310049 \)

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:  \(934801626748320922851562500000000000000000000=2^{20}\cdot 3^{10}\cdot 5^{34}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $177.23$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3300=2^{2}\cdot 3\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3300}(1,·)$, $\chi_{3300}(131,·)$, $\chi_{3300}(769,·)$, $\chi_{3300}(1739,·)$, $\chi_{3300}(2641,·)$, $\chi_{3300}(2771,·)$, $\chi_{3300}(1429,·)$, $\chi_{3300}(791,·)$, $\chi_{3300}(1079,·)$, $\chi_{3300}(2399,·)$, $\chi_{3300}(419,·)$, $\chi_{3300}(1321,·)$, $\chi_{3300}(1451,·)$, $\chi_{3300}(109,·)$, $\chi_{3300}(2749,·)$, $\chi_{3300}(3059,·)$, $\chi_{3300}(2089,·)$, $\chi_{3300}(2111,·)$, $\chi_{3300}(1981,·)$, $\chi_{3300}(661,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{525} a^{10} - \frac{1}{15} a^{8} - \frac{2}{105} a^{7} + \frac{1}{5} a^{6} - \frac{1}{75} a^{5} + \frac{46}{105} a^{4} - \frac{4}{15} a^{3} + \frac{1}{15} a^{2} + \frac{2}{105} a - \frac{26}{75}$, $\frac{1}{525} a^{11} - \frac{1}{15} a^{9} - \frac{2}{105} a^{8} + \frac{2}{35} a^{7} - \frac{1}{75} a^{6} + \frac{46}{105} a^{5} - \frac{4}{15} a^{4} + \frac{1}{15} a^{3} + \frac{2}{105} a^{2} - \frac{107}{525} a$, $\frac{1}{525} a^{12} - \frac{2}{105} a^{9} + \frac{1}{105} a^{8} + \frac{6}{175} a^{7} + \frac{46}{105} a^{6} + \frac{4}{15} a^{5} + \frac{2}{5} a^{4} - \frac{11}{35} a^{3} - \frac{82}{525} a^{2} - \frac{1}{21} a - \frac{2}{15}$, $\frac{1}{525} a^{13} + \frac{1}{105} a^{9} - \frac{32}{525} a^{8} - \frac{4}{105} a^{7} + \frac{4}{15} a^{6} + \frac{4}{15} a^{5} + \frac{1}{15} a^{4} + \frac{31}{175} a^{3} + \frac{1}{21} a^{2} + \frac{12}{35} a - \frac{7}{15}$, $\frac{1}{3675} a^{14} - \frac{26}{525} a^{9} + \frac{1}{735} a^{8} - \frac{1}{105} a^{7} + \frac{7}{15} a^{6} - \frac{4}{15} a^{5} + \frac{32}{75} a^{4} - \frac{4}{21} a^{3} + \frac{136}{735} a^{2} + \frac{43}{105} a - \frac{7}{15}$, $\frac{1}{14700} a^{15} + \frac{1}{2100} a^{13} - \frac{1}{2100} a^{12} - \frac{1}{1050} a^{11} - \frac{1}{1050} a^{10} + \frac{1}{196} a^{9} + \frac{113}{2100} a^{8} - \frac{31}{700} a^{7} + \frac{243}{700} a^{6} + \frac{121}{2100} a^{5} + \frac{8}{21} a^{4} + \frac{2363}{7350} a^{3} - \frac{179}{1050} a^{2} + \frac{69}{350} a - \frac{49}{300}$, $\frac{1}{14700} a^{16} - \frac{1}{14700} a^{14} - \frac{1}{2100} a^{13} - \frac{1}{1050} a^{12} - \frac{1}{1050} a^{11} - \frac{3}{4900} a^{10} + \frac{1}{100} a^{9} + \frac{149}{14700} a^{8} - \frac{11}{2100} a^{7} - \frac{333}{700} a^{6} - \frac{8}{175} a^{5} + \frac{377}{2450} a^{4} + \frac{23}{150} a^{3} - \frac{1691}{7350} a^{2} + \frac{817}{2100} a - \frac{2}{75}$, $\frac{1}{632100} a^{17} - \frac{1}{105350} a^{16} - \frac{1}{210700} a^{15} + \frac{17}{210700} a^{14} - \frac{29}{45150} a^{13} + \frac{1}{1075} a^{12} + \frac{523}{632100} a^{11} + \frac{341}{632100} a^{10} - \frac{6529}{210700} a^{9} + \frac{13049}{210700} a^{8} + \frac{743}{30100} a^{7} - \frac{31}{129} a^{6} + \frac{30004}{158025} a^{5} + \frac{14459}{316050} a^{4} - \frac{6143}{105350} a^{3} + \frac{2503}{25284} a^{2} - \frac{11533}{45150} a - \frac{251}{1290}$, $\frac{1}{220522889696206862100} a^{18} - \frac{6591794967976}{18376907474683905175} a^{17} + \frac{8322586924292}{375038928054773575} a^{16} - \frac{3170900084228249}{220522889696206862100} a^{15} + \frac{3799690240865297}{220522889696206862100} a^{14} + \frac{179510164531729}{900093427331456580} a^{13} - \frac{67838492554231457}{220522889696206862100} a^{12} + \frac{161309144622541933}{220522889696206862100} a^{11} - \frac{2072380498978473}{2625272496383415025} a^{10} - \frac{3477549727596431206}{55130722424051715525} a^{9} + \frac{1026842208396002359}{36753814949367810350} a^{8} - \frac{2054072267029044917}{31503269956600980300} a^{7} + \frac{105977259158474706283}{220522889696206862100} a^{6} + \frac{2545827622187773279}{18376907474683905175} a^{5} + \frac{427627249284970929}{2625272496383415025} a^{4} + \frac{10649224647227745107}{73507629898735620700} a^{3} - \frac{52309950861725097239}{110261444848103431050} a^{2} - \frac{467270706783513909}{2100217997106732020} a - \frac{213964227716510933}{450046713665728290}$, $\frac{1}{8879441365840050101763886483195028894694813302100} a^{19} - \frac{1059746577840374680832029233}{1479906894306675016960647747199171482449135550350} a^{18} - \frac{580659327343143224744206669425486908703979}{1268491623691435728823412354742146984956401900300} a^{17} - \frac{20147830166824117622298947410146380110176853}{2219860341460012525440971620798757223673703325525} a^{16} + \frac{66444190210327674474500106950685438395573642}{2219860341460012525440971620798757223673703325525} a^{15} + \frac{1230830875107718959278324336909820796953679}{9833268400708804099406297323582534767103890700} a^{14} - \frac{145339104320633923735703753776695692432140857}{739953447153337508480323873599585741224567775175} a^{13} - \frac{6066928487419246262270477531179784411980533361}{8879441365840050101763886483195028894694813302100} a^{12} + \frac{57297773836271767930938965678379504528604401}{84566108246095715254894156982809798997093460020} a^{11} + \frac{247932092170629555773179592623972708140059671}{739953447153337508480323873599585741224567775175} a^{10} + \frac{680699100095629207094453119207825958215678592}{29598137886133500339212954943983429648982711007} a^{9} - \frac{29085635176916633334924316022051236446862161509}{422830541230478576274470784914048994985467300100} a^{8} - \frac{31946208216922748395985007967700836725561866589}{591962757722670006784259098879668592979654220140} a^{7} + \frac{67229235570757506691083068381562805109662965779}{2959813788613350033921295494398342964898271100700} a^{6} - \frac{6804318072565037679799956932940971279658825173}{634245811845717864411706177371073492478200950150} a^{5} - \frac{2740086225165124073972783900835065083375971515483}{8879441365840050101763886483195028894694813302100} a^{4} - \frac{189879326029755598695796590599588072946978385414}{443972068292002505088194324159751444734740665105} a^{3} - \frac{267230957419713613038716052940805076691668432439}{634245811845717864411706177371073492478200950150} a^{2} - \frac{3455417015169831405648445990932678579566538799}{25887584156968076098436986831472387448089834700} a - \frac{1334720744623296988095833465937886276749114008}{6471896039242019024609246707868096862022458675}$

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

Class group and class number

$C_{2}\times C_{7719688}$, which has order $15439376$ (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:  \( -1 \) (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:  \( 180801817.57689384 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-33}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{15}, \sqrt{-33})\), 5.5.390625.1, 10.0.6114905156250000000000.1, 10.10.189843750000000000.1, 10.0.122872161865234375.1

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

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 ${\href{/LocalNumberField/7.1.0.1}{1} }^{20}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
3Data not computed
5Data not computed
$11$11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$