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Properties

Label 20.0.78125000000...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{24}\cdot 5^{31}$
Root discriminant $27.84$
Ramified primes $2, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_5:F_5$ (as 20T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![400, 1000, 1000, 1000, 1625, 850, -1275, -750, 1925, 750, -1860, -500, 1025, 150, -350, -20, 80, 0, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^18 + 80*x^16 - 20*x^15 - 350*x^14 + 150*x^13 + 1025*x^12 - 500*x^11 - 1860*x^10 + 750*x^9 + 1925*x^8 - 750*x^7 - 1275*x^6 + 850*x^5 + 1625*x^4 + 1000*x^3 + 1000*x^2 + 1000*x + 400)
 
gp: K = bnfinit(x^20 - 10*x^18 + 80*x^16 - 20*x^15 - 350*x^14 + 150*x^13 + 1025*x^12 - 500*x^11 - 1860*x^10 + 750*x^9 + 1925*x^8 - 750*x^7 - 1275*x^6 + 850*x^5 + 1625*x^4 + 1000*x^3 + 1000*x^2 + 1000*x + 400, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{18} + 80 x^{16} - 20 x^{15} - 350 x^{14} + 150 x^{13} + 1025 x^{12} - 500 x^{11} - 1860 x^{10} + 750 x^{9} + 1925 x^{8} - 750 x^{7} - 1275 x^{6} + 850 x^{5} + 1625 x^{4} + 1000 x^{3} + 1000 x^{2} + 1000 x + 400 \)

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:  \(78125000000000000000000000000=2^{24}\cdot 5^{31}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.84$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10}$, $\frac{1}{5} a^{11}$, $\frac{1}{25} a^{12} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{2}$, $\frac{1}{25} a^{13} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{3}$, $\frac{1}{25} a^{14} - \frac{2}{5} a^{9} - \frac{1}{5} a^{4}$, $\frac{1}{25} a^{15} - \frac{1}{5} a^{5}$, $\frac{1}{25} a^{16} - \frac{1}{5} a^{6}$, $\frac{1}{150} a^{17} + \frac{1}{75} a^{15} + \frac{1}{75} a^{14} + \frac{1}{75} a^{13} + \frac{1}{75} a^{12} - \frac{1}{15} a^{11} + \frac{1}{15} a^{10} - \frac{1}{10} a^{9} - \frac{4}{15} a^{8} + \frac{1}{3} a^{7} + \frac{1}{10} a^{5} + \frac{4}{15} a^{4} + \frac{1}{10} a^{3} + \frac{4}{15} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{900} a^{18} - \frac{1}{450} a^{17} + \frac{1}{450} a^{16} + \frac{4}{225} a^{15} - \frac{2}{225} a^{14} + \frac{4}{225} a^{13} - \frac{7}{450} a^{12} - \frac{1}{10} a^{11} + \frac{17}{180} a^{10} - \frac{13}{90} a^{9} - \frac{2}{9} a^{8} - \frac{5}{18} a^{7} - \frac{29}{60} a^{6} - \frac{4}{45} a^{5} + \frac{53}{180} a^{4} + \frac{11}{45} a^{3} + \frac{49}{180} a^{2} - \frac{5}{18} a + \frac{4}{9}$, $\frac{1}{147449666863764545400} a^{19} + \frac{1423223430386944}{18431208357970568175} a^{18} - \frac{1323715679196545}{982997779091763636} a^{17} + \frac{55764432020642186}{6143736119323522725} a^{16} + \frac{3167105576327861}{259594483915078425} a^{15} - \frac{96001636241985247}{36862416715941136350} a^{14} + \frac{351418986231888829}{73724833431882272700} a^{13} - \frac{249805194813740249}{14744966686376454540} a^{12} - \frac{425368332143474879}{5897986674550581816} a^{11} - \frac{234707693333152049}{2457494447729409090} a^{10} - \frac{36358770906268714}{1228747223864704545} a^{9} - \frac{1960388529398019403}{4914988895458818180} a^{8} - \frac{2347863403843493083}{5897986674550581816} a^{7} - \frac{856271405809337987}{14744966686376454540} a^{6} - \frac{4163997741215606483}{29489933372752909080} a^{5} + \frac{3249697375117650389}{14744966686376454540} a^{4} + \frac{4848866620592424779}{9829977790917636360} a^{3} + \frac{712138323702383}{737248334318822727} a^{2} + \frac{74127834168425875}{163832963181960606} a - \frac{339153628479900718}{737248334318822727}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( -\frac{458337830400175493}{147449666863764545400} a^{19} + \frac{19928739521897713}{18431208357970568175} a^{18} + \frac{153228506499709907}{4914988895458818180} a^{17} - \frac{70063120096258939}{6143736119323522725} a^{16} - \frac{2585534954653426}{10383779356603137} a^{15} + \frac{5673006145274857661}{36862416715941136350} a^{14} + \frac{15746645405116338527}{14744966686376454540} a^{13} - \frac{13072327530932902187}{14744966686376454540} a^{12} - \frac{88526434627567624561}{29489933372752909080} a^{11} + \frac{6916096223800292551}{2457494447729409090} a^{10} + \frac{2503053063428332169}{491498889545881818} a^{9} - \frac{22874453939617245049}{4914988895458818180} a^{8} - \frac{28312999276266403441}{5897986674550581816} a^{7} + \frac{68794176305767512343}{14744966686376454540} a^{6} + \frac{17177327425266448151}{5897986674550581816} a^{5} - \frac{59021910871291106677}{14744966686376454540} a^{4} - \frac{8963601184051215599}{1965995558183527272} a^{3} - \frac{750687113854158856}{737248334318822727} a^{2} - \frac{160945336488067336}{81916481590980303} a - \frac{1410664807717005592}{737248334318822727} \) (order $10$)
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:  \( 11302864.7053 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_5:F_5$ (as 20T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 20 conjugacy class representatives for $C_2\times C_5:F_5$
Character table for $C_2\times C_5:F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.2.125000000000000.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
5Data not computed