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Properties

Label 20.0.18512297918...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{38}\cdot 5^{22}\cdot 7^{10}$
Root discriminant $57.99$
Ramified primes $2, 5, 7$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1134035519, -25546630, 71093645, -116976690, 127060815, -93178236, 49412670, -16134180, 4537935, -336910, 67097, 770, 45, -540, 90, 228, -105, -30, 35, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 35*x^18 - 30*x^17 - 105*x^16 + 228*x^15 + 90*x^14 - 540*x^13 + 45*x^12 + 770*x^11 + 67097*x^10 - 336910*x^9 + 4537935*x^8 - 16134180*x^7 + 49412670*x^6 - 93178236*x^5 + 127060815*x^4 - 116976690*x^3 + 71093645*x^2 - 25546630*x + 1134035519)
 
gp: K = bnfinit(x^20 - 10*x^19 + 35*x^18 - 30*x^17 - 105*x^16 + 228*x^15 + 90*x^14 - 540*x^13 + 45*x^12 + 770*x^11 + 67097*x^10 - 336910*x^9 + 4537935*x^8 - 16134180*x^7 + 49412670*x^6 - 93178236*x^5 + 127060815*x^4 - 116976690*x^3 + 71093645*x^2 - 25546630*x + 1134035519, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 35 x^{18} - 30 x^{17} - 105 x^{16} + 228 x^{15} + 90 x^{14} - 540 x^{13} + 45 x^{12} + 770 x^{11} + 67097 x^{10} - 336910 x^{9} + 4537935 x^{8} - 16134180 x^{7} + 49412670 x^{6} - 93178236 x^{5} + 127060815 x^{4} - 116976690 x^{3} + 71093645 x^{2} - 25546630 x + 1134035519 \)

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:  \(185122979184640000000000000000000000=2^{38}\cdot 5^{22}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.99$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{1}{7} a^{2} + \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{5} + \frac{2}{7} a^{3} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{6} - \frac{3}{7} a^{3} - \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{7} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{49} a^{8} + \frac{3}{49} a^{7} + \frac{2}{49} a^{6} + \frac{1}{49} a^{5} + \frac{2}{49} a^{4} + \frac{20}{49} a^{3} + \frac{9}{49} a^{2} + \frac{11}{49} a + \frac{15}{49}$, $\frac{1}{49} a^{9} + \frac{2}{49} a^{6} - \frac{1}{49} a^{5} + \frac{12}{49} a^{3} + \frac{12}{49} a^{2} + \frac{17}{49} a - \frac{3}{49}$, $\frac{1}{245} a^{10} - \frac{1}{49} a^{7} - \frac{3}{49} a^{6} + \frac{2}{35} a^{5} + \frac{1}{49} a^{4} + \frac{8}{49} a^{3} + \frac{2}{49} a^{2} - \frac{9}{49} a + \frac{4}{35}$, $\frac{1}{245} a^{11} - \frac{11}{245} a^{6} + \frac{2}{49} a^{5} + \frac{3}{49} a^{4} + \frac{8}{49} a^{3} + \frac{1}{7} a^{2} + \frac{83}{245} a + \frac{22}{49}$, $\frac{1}{1715} a^{12} + \frac{1}{1715} a^{11} + \frac{2}{1715} a^{10} + \frac{2}{343} a^{9} + \frac{1}{343} a^{8} - \frac{111}{1715} a^{7} - \frac{106}{1715} a^{6} + \frac{13}{1715} a^{5} + \frac{22}{343} a^{4} - \frac{107}{343} a^{3} - \frac{677}{1715} a^{2} - \frac{232}{1715} a - \frac{174}{1715}$, $\frac{1}{1715} a^{13} + \frac{1}{1715} a^{11} + \frac{1}{1715} a^{10} - \frac{1}{343} a^{9} - \frac{11}{1715} a^{8} + \frac{22}{343} a^{7} - \frac{8}{245} a^{6} + \frac{104}{1715} a^{5} + \frac{4}{343} a^{4} + \frac{208}{1715} a^{3} + \frac{68}{343} a^{2} + \frac{793}{1715} a + \frac{573}{1715}$, $\frac{1}{1715} a^{14} + \frac{2}{245} a^{9} - \frac{10}{343} a^{7} - \frac{1}{49} a^{6} - \frac{1}{49} a^{5} - \frac{11}{245} a^{4} - \frac{8}{49} a^{3} - \frac{6}{49} a^{2} - \frac{23}{49} a - \frac{115}{343}$, $\frac{1}{1715} a^{15} - \frac{3}{343} a^{8} - \frac{3}{49} a^{7} + \frac{1}{245} a^{5} - \frac{1}{49} a^{4} + \frac{12}{49} a^{3} + \frac{10}{49} a^{2} - \frac{10}{343} a + \frac{89}{245}$, $\frac{1}{60025} a^{16} - \frac{8}{60025} a^{15} - \frac{8}{60025} a^{14} - \frac{2}{8575} a^{13} - \frac{2}{8575} a^{12} - \frac{2}{1225} a^{11} - \frac{1}{1715} a^{10} - \frac{127}{60025} a^{9} + \frac{309}{60025} a^{8} + \frac{134}{60025} a^{7} + \frac{66}{1715} a^{6} - \frac{59}{1225} a^{5} + \frac{298}{8575} a^{4} + \frac{2059}{8575} a^{3} - \frac{25012}{60025} a^{2} - \frac{19186}{60025} a + \frac{5636}{60025}$, $\frac{1}{60025} a^{17} - \frac{2}{60025} a^{15} - \frac{8}{60025} a^{14} + \frac{2}{8575} a^{13} + \frac{3}{8575} a^{11} - \frac{92}{60025} a^{10} + \frac{64}{8575} a^{9} - \frac{159}{60025} a^{8} - \frac{678}{60025} a^{7} + \frac{307}{8575} a^{6} - \frac{606}{8575} a^{5} + \frac{198}{8575} a^{4} - \frac{29688}{60025} a^{3} + \frac{1539}{8575} a^{2} + \frac{26343}{60025} a + \frac{12993}{60025}$, $\frac{1}{935857098061423066088525} a^{18} - \frac{9}{935857098061423066088525} a^{17} - \frac{7279500773003928317}{935857098061423066088525} a^{16} + \frac{11647201236806285348}{187171419612284613217705} a^{15} - \frac{247279341802477457744}{935857098061423066088525} a^{14} + \frac{23733716341447365087}{133693871151631866584075} a^{13} + \frac{1276356656114251968}{133693871151631866584075} a^{12} + \frac{436363114744921113749}{935857098061423066088525} a^{11} - \frac{1086748407371728466664}{935857098061423066088525} a^{10} + \frac{90598901084571211964}{935857098061423066088525} a^{9} + \frac{3195799187860687459293}{935857098061423066088525} a^{8} - \frac{46400201433467153092704}{935857098061423066088525} a^{7} - \frac{4875429949076607569449}{133693871151631866584075} a^{6} + \frac{9418061657325107873502}{133693871151631866584075} a^{5} + \frac{43445066994466782356123}{935857098061423066088525} a^{4} + \frac{58804136778163498629599}{187171419612284613217705} a^{3} - \frac{254912877820105140875524}{935857098061423066088525} a^{2} - \frac{315580336293352061633474}{935857098061423066088525} a - \frac{39597196854686960234867}{935857098061423066088525}$, $\frac{1}{32215814659529797089225198295025} a^{19} + \frac{17211921}{32215814659529797089225198295025} a^{18} + \frac{11694532067395490092094964}{4602259237075685298460742613575} a^{17} - \frac{944646737457764449029098}{1895047921148811593483835193825} a^{16} - \frac{8399314488412066474171164846}{32215814659529797089225198295025} a^{15} + \frac{6644046190982292345362380402}{32215814659529797089225198295025} a^{14} - \frac{21427722218899757598772264}{131493121059305294241735503245} a^{13} + \frac{2809905644280111432156287983}{32215814659529797089225198295025} a^{12} + \frac{31565247657696918963769629729}{32215814659529797089225198295025} a^{11} + \frac{1843294295053589283495298227}{4602259237075685298460742613575} a^{10} - \frac{45495937126580584774790542368}{6443162931905959417845039659005} a^{9} - \frac{319066083804354048632550688283}{32215814659529797089225198295025} a^{8} + \frac{1733819738290873544439435290783}{32215814659529797089225198295025} a^{7} + \frac{131010059732666503891963566732}{4602259237075685298460742613575} a^{6} + \frac{1155950484926306789093102577554}{32215814659529797089225198295025} a^{5} + \frac{2218966895257557628753858635264}{32215814659529797089225198295025} a^{4} + \frac{19878110502217325607788449024}{4602259237075685298460742613575} a^{3} + \frac{2699881652271503044853366031543}{32215814659529797089225198295025} a^{2} + \frac{6987109273271187518117068396214}{32215814659529797089225198295025} a - \frac{2999959851101414761153305171521}{32215814659529797089225198295025}$

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

$C_2\times F_5$ (as 20T13):

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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{5}, \sqrt{-14})\), 5.1.50000.1, 10.2.12500000000.1, 10.0.86051840000000000.4, 10.0.430259200000000000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$5$5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$