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Properties

Label 20.0.19331544372...3125.2
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 11^{16}\cdot 13^{10}$
Root discriminant $82.09$
Ramified primes $5, 11, 13$
Class number $10$ (GRH)
Class group $[10]$ (GRH)
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121715395, -464217385, 502243165, -184199435, 380740776, -123999347, 141614400, -45897320, 39544565, -8576736, 8039592, -972755, 1053075, -80690, 84557, -5019, 3990, -190, 100, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 100*x^18 - 190*x^17 + 3990*x^16 - 5019*x^15 + 84557*x^14 - 80690*x^13 + 1053075*x^12 - 972755*x^11 + 8039592*x^10 - 8576736*x^9 + 39544565*x^8 - 45897320*x^7 + 141614400*x^6 - 123999347*x^5 + 380740776*x^4 - 184199435*x^3 + 502243165*x^2 - 464217385*x + 121715395)
 
gp: K = bnfinit(x^20 - 3*x^19 + 100*x^18 - 190*x^17 + 3990*x^16 - 5019*x^15 + 84557*x^14 - 80690*x^13 + 1053075*x^12 - 972755*x^11 + 8039592*x^10 - 8576736*x^9 + 39544565*x^8 - 45897320*x^7 + 141614400*x^6 - 123999347*x^5 + 380740776*x^4 - 184199435*x^3 + 502243165*x^2 - 464217385*x + 121715395, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 100 x^{18} - 190 x^{17} + 3990 x^{16} - 5019 x^{15} + 84557 x^{14} - 80690 x^{13} + 1053075 x^{12} - 972755 x^{11} + 8039592 x^{10} - 8576736 x^{9} + 39544565 x^{8} - 45897320 x^{7} + 141614400 x^{6} - 123999347 x^{5} + 380740776 x^{4} - 184199435 x^{3} + 502243165 x^{2} - 464217385 x + 121715395 \)

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:  \(193315443721344440894830801055908203125=5^{15}\cdot 11^{16}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.09$
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:  $5, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{4}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{5}$, $\frac{1}{10} a^{11} - \frac{1}{10} a^{10} - \frac{1}{10} a^{9} - \frac{1}{10} a^{8} + \frac{1}{5} a^{7} + \frac{3}{10} a^{6} - \frac{2}{5} a^{5} - \frac{3}{10} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{50} a^{12} + \frac{1}{50} a^{11} - \frac{3}{50} a^{10} + \frac{3}{50} a^{9} - \frac{23}{50} a^{7} - \frac{9}{25} a^{6} - \frac{1}{50} a^{5} + \frac{1}{50} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{3}{10} a + \frac{2}{5}$, $\frac{1}{50} a^{13} + \frac{1}{50} a^{11} + \frac{1}{50} a^{10} + \frac{1}{25} a^{9} + \frac{1}{25} a^{8} + \frac{1}{10} a^{7} + \frac{1}{25} a^{6} - \frac{4}{25} a^{5} + \frac{12}{25} a^{4} - \frac{1}{10} a^{3} - \frac{1}{2} a^{2} - \frac{3}{10} a + \frac{1}{10}$, $\frac{1}{150} a^{14} - \frac{1}{150} a^{12} + \frac{2}{75} a^{11} + \frac{1}{50} a^{10} + \frac{1}{150} a^{9} - \frac{1}{15} a^{8} + \frac{14}{75} a^{7} - \frac{47}{150} a^{6} + \frac{6}{25} a^{5} - \frac{31}{75} a^{4} - \frac{1}{5} a^{3} - \frac{7}{30} a^{2} - \frac{13}{30} a - \frac{13}{30}$, $\frac{1}{2850} a^{15} - \frac{1}{570} a^{14} - \frac{1}{285} a^{13} + \frac{4}{475} a^{12} - \frac{101}{2850} a^{11} - \frac{1}{75} a^{10} + \frac{22}{475} a^{9} - \frac{9}{95} a^{8} + \frac{833}{2850} a^{7} + \frac{883}{2850} a^{6} - \frac{157}{570} a^{5} - \frac{11}{2850} a^{4} + \frac{61}{285} a^{3} + \frac{97}{570} a^{2} + \frac{32}{285} a + \frac{13}{285}$, $\frac{1}{14250} a^{16} - \frac{1}{14250} a^{15} - \frac{11}{14250} a^{14} - \frac{13}{1425} a^{13} + \frac{3}{475} a^{12} - \frac{27}{4750} a^{11} - \frac{352}{7125} a^{10} + \frac{53}{7125} a^{9} - \frac{2}{75} a^{8} - \frac{943}{2850} a^{7} - \frac{997}{4750} a^{6} + \frac{6881}{14250} a^{5} + \frac{727}{4750} a^{4} - \frac{178}{475} a^{3} - \frac{1163}{2850} a^{2} + \frac{7}{570} a - \frac{328}{1425}$, $\frac{1}{14250} a^{17} - \frac{1}{7125} a^{15} - \frac{1}{14250} a^{14} - \frac{14}{1425} a^{13} + \frac{59}{14250} a^{12} + \frac{13}{475} a^{11} + \frac{339}{4750} a^{10} - \frac{63}{4750} a^{9} + \frac{28}{1425} a^{8} + \frac{122}{7125} a^{7} - \frac{667}{2850} a^{6} + \frac{392}{2375} a^{5} + \frac{3163}{7125} a^{4} - \frac{121}{475} a^{3} + \frac{227}{475} a^{2} + \frac{7}{50} a + \frac{1409}{2850}$, $\frac{1}{5757000} a^{18} + \frac{59}{1919000} a^{17} + \frac{3}{383800} a^{16} - \frac{397}{5757000} a^{15} + \frac{851}{5757000} a^{14} + \frac{4583}{719625} a^{13} - \frac{611}{479750} a^{12} - \frac{27541}{575700} a^{11} + \frac{513907}{5757000} a^{10} + \frac{5847}{101000} a^{9} + \frac{73753}{1919000} a^{8} + \frac{2194183}{5757000} a^{7} + \frac{84649}{287850} a^{6} - \frac{2768233}{5757000} a^{5} + \frac{34451}{1439250} a^{4} - \frac{94199}{287850} a^{3} - \frac{112277}{287850} a^{2} - \frac{45703}{1151400} a - \frac{134013}{383800}$, $\frac{1}{6047312287748171833144409335765547799934851504192894000} a^{19} - \frac{20058105780833756014022766260008574588686823336}{377957017984260739571525583485346737495928219012055875} a^{18} + \frac{3701976112103507392267230400271352926325419642831}{755914035968521479143051166970693474991856438024111750} a^{17} - \frac{31523345890739655082865521878465665613421318907947}{1007885381291361972190734889294257966655808584032149000} a^{16} + \frac{23962268679793913836530179848784586336409873225571}{1511828071937042958286102333941386949983712876048223500} a^{15} - \frac{15027031507193341357388623358303557767340339563103763}{6047312287748171833144409335765547799934851504192894000} a^{14} + \frac{2053058076174285715192353973642637248678121804002899}{251971345322840493047683722323564491663952146008037250} a^{13} - \frac{6509220356100477417705540786887086564851834195960409}{3023656143874085916572204667882773899967425752096447000} a^{12} - \frac{253833218901261759630484851415943368001121695286096803}{6047312287748171833144409335765547799934851504192894000} a^{11} - \frac{676079635323163211883616741716390929614980318837453}{26523299507667420320808812876164683333047594316635500} a^{10} + \frac{11627077989934447555397650379376371666152083164899786}{377957017984260739571525583485346737495928219012055875} a^{9} + \frac{76888739787921347990454994233568147381325039883777}{17377334160195896072254049815418240804410492828140500} a^{8} + \frac{1430873183302661005713264798997032550977423301143952357}{6047312287748171833144409335765547799934851504192894000} a^{7} - \frac{296075044354648410780167452636586347294061439480654271}{2015770762582723944381469778588515933311617168064298000} a^{6} + \frac{88040954061221695485641995245374624364748645832869441}{6047312287748171833144409335765547799934851504192894000} a^{5} - \frac{320906182949085520868173444039744238850377194550956847}{1511828071937042958286102333941386949983712876048223500} a^{4} + \frac{59646566145218980337561654710497594084925327588441819}{302365614387408591657220466788277389996742575209644700} a^{3} - \frac{28730052731213599501895647835096709364974744108958051}{1209462457549634366628881867153109559986970300838578800} a^{2} + \frac{10617551739246207792250888296591074213576961313304171}{75591403596852147914305116697069347499185643802411175} a - \frac{10240222485429746678203212618804477334148913432767563}{63655918818401808769941150902795239999314226359925200}$

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

Class group and class number

$C_{10}$, which has order $10$ (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:  \( 11176801552.71916 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.21125.1, 5.1.1830125.1, 10.2.16746787578125.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.10.8.1$x^{10} + 220 x^{5} + 41503$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.1$x^{10} + 220 x^{5} + 41503$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$