↑

Properties

Label 20.0.12719384508...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{10}\cdot 5^{22}\cdot 13^{10}$
Root discriminant $63.86$
Ramified primes $2, 3, 5, 13$
Class number $20$ (GRH)
Class group $[2, 10]$ (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1118589, 3238350, 4384705, 2020440, -1070835, -1164726, -87670, 450680, 257955, -166250, -98105, 19180, 28315, -1520, -4550, -88, 555, 0, -35, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 35*x^18 + 555*x^16 - 88*x^15 - 4550*x^14 - 1520*x^13 + 28315*x^12 + 19180*x^11 - 98105*x^10 - 166250*x^9 + 257955*x^8 + 450680*x^7 - 87670*x^6 - 1164726*x^5 - 1070835*x^4 + 2020440*x^3 + 4384705*x^2 + 3238350*x + 1118589)
 
gp: K = bnfinit(x^20 - 35*x^18 + 555*x^16 - 88*x^15 - 4550*x^14 - 1520*x^13 + 28315*x^12 + 19180*x^11 - 98105*x^10 - 166250*x^9 + 257955*x^8 + 450680*x^7 - 87670*x^6 - 1164726*x^5 - 1070835*x^4 + 2020440*x^3 + 4384705*x^2 + 3238350*x + 1118589, 1)
 

Normalized defining polynomial

\( x^{20} - 35 x^{18} + 555 x^{16} - 88 x^{15} - 4550 x^{14} - 1520 x^{13} + 28315 x^{12} + 19180 x^{11} - 98105 x^{10} - 166250 x^{9} + 257955 x^{8} + 450680 x^{7} - 87670 x^{6} - 1164726 x^{5} - 1070835 x^{4} + 2020440 x^{3} + 4384705 x^{2} + 3238350 x + 1118589 \)

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:  \(1271938450811187656250000000000000000=2^{16}\cdot 3^{10}\cdot 5^{22}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.86$
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:  $2, 3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{13} + \frac{1}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{8} a - \frac{3}{8}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} + \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{3}{8} a^{6} + \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{12} - \frac{1}{4} a^{10} + \frac{1}{16} a^{8} + \frac{1}{4} a^{7} - \frac{1}{8} a^{6} - \frac{3}{8} a^{5} - \frac{1}{2} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{8} a + \frac{1}{16}$, $\frac{1}{16} a^{17} - \frac{1}{8} a^{13} - \frac{1}{4} a^{11} + \frac{1}{16} a^{9} + \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{3}{8} a^{6} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{16} a$, $\frac{1}{1164512} a^{18} - \frac{14697}{1164512} a^{17} - \frac{11407}{1164512} a^{16} - \frac{2036}{36391} a^{15} - \frac{18775}{582256} a^{14} + \frac{50267}{582256} a^{13} + \frac{36161}{582256} a^{12} - \frac{4747}{36391} a^{11} + \frac{120185}{1164512} a^{10} + \frac{473827}{1164512} a^{9} - \frac{247265}{1164512} a^{8} - \frac{133281}{291128} a^{7} + \frac{16686}{36391} a^{6} - \frac{106497}{291128} a^{5} + \frac{23643}{582256} a^{4} + \frac{16277}{72782} a^{3} - \frac{191217}{1164512} a^{2} - \frac{291039}{1164512} a + \frac{570109}{1164512}$, $\frac{1}{351357377256869237602857214348070695155734890887860384} a^{19} - \frac{30969764623095638403806486685958458462381757615}{175678688628434618801428607174035347577867445443930192} a^{18} + \frac{1146575822481939006583356306108181312495600575551495}{43919672157108654700357151793508836894466861360982548} a^{17} - \frac{8997766232736473307973865563948927732618141252770091}{351357377256869237602857214348070695155734890887860384} a^{16} + \frac{2047598397000206305178600375689332790031735509215965}{175678688628434618801428607174035347577867445443930192} a^{15} + \frac{547544352730661372607687019930744534609974384313376}{10979918039277163675089287948377209223616715340245637} a^{14} - \frac{133124370642646299182453296321222654214908008012847}{87839344314217309400714303587017673788933722721965096} a^{13} + \frac{9437725473723195683628549131629421706288485813778097}{175678688628434618801428607174035347577867445443930192} a^{12} + \frac{68483176761495706383994044998193677822874658921774437}{351357377256869237602857214348070695155734890887860384} a^{11} - \frac{15121767924635600591984004088279722264134936463161735}{175678688628434618801428607174035347577867445443930192} a^{10} + \frac{16410947298251559488055330410567009906604319781448621}{175678688628434618801428607174035347577867445443930192} a^{9} - \frac{72957208846741368693310738513570302468794343178746873}{351357377256869237602857214348070695155734890887860384} a^{8} - \frac{42539726659091281117266955541382487298403361076318173}{87839344314217309400714303587017673788933722721965096} a^{7} + \frac{18455034957780011383357415860998891504250419961947177}{43919672157108654700357151793508836894466861360982548} a^{6} - \frac{79153470483949680052027313656670806962185601172143773}{175678688628434618801428607174035347577867445443930192} a^{5} + \frac{31140589476831959872022874232089544491174855395627529}{175678688628434618801428607174035347577867445443930192} a^{4} + \frac{118178320522085080707000072373088141979132276506150591}{351357377256869237602857214348070695155734890887860384} a^{3} + \frac{25947032385289499011779176242573883470650387420828889}{175678688628434618801428607174035347577867445443930192} a^{2} + \frac{41285798263371457703515503687969164662804531773603433}{175678688628434618801428607174035347577867445443930192} a - \frac{108466504079499765283663483983668550926298472267086947}{351357377256869237602857214348070695155734890887860384}$

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

Class group and class number

$C_{2}\times C_{10}$, which has order $20$ (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:  \( 1065874435.9547355 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-195}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{13}, \sqrt{-15})\), 5.1.50000.1, 10.0.3037500000000.2, 10.0.1127802487500000000.1, 10.2.928232500000000.3

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$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}$