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Properties

Label 20.0.37145828047...8229.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{16}\cdot 29^{15}$
Root discriminant $30.10$
Ramified primes $3, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2075, -1020, 6579, -2026, 5813, -1452, 2414, -413, 986, -323, 852, -264, 317, -158, 63, -28, 54, -11, -2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 2*x^18 - 11*x^17 + 54*x^16 - 28*x^15 + 63*x^14 - 158*x^13 + 317*x^12 - 264*x^11 + 852*x^10 - 323*x^9 + 986*x^8 - 413*x^7 + 2414*x^6 - 1452*x^5 + 5813*x^4 - 2026*x^3 + 6579*x^2 - 1020*x + 2075)
 
gp: K = bnfinit(x^20 - x^19 - 2*x^18 - 11*x^17 + 54*x^16 - 28*x^15 + 63*x^14 - 158*x^13 + 317*x^12 - 264*x^11 + 852*x^10 - 323*x^9 + 986*x^8 - 413*x^7 + 2414*x^6 - 1452*x^5 + 5813*x^4 - 2026*x^3 + 6579*x^2 - 1020*x + 2075, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 2 x^{18} - 11 x^{17} + 54 x^{16} - 28 x^{15} + 63 x^{14} - 158 x^{13} + 317 x^{12} - 264 x^{11} + 852 x^{10} - 323 x^{9} + 986 x^{8} - 413 x^{7} + 2414 x^{6} - 1452 x^{5} + 5813 x^{4} - 2026 x^{3} + 6579 x^{2} - 1020 x + 2075 \)

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:  \(371458280474198465736072578229=3^{16}\cdot 29^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.10$
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:  $3, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{11} + \frac{1}{10} a^{10} + \frac{1}{5} a^{9} - \frac{1}{2} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{3}{10} a^{5} + \frac{1}{10} a^{3} - \frac{2}{5} a^{2} - \frac{3}{10} a$, $\frac{1}{10} a^{12} + \frac{1}{10} a^{10} - \frac{1}{5} a^{9} + \frac{1}{10} a^{8} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{2} a^{3} + \frac{1}{10} a^{2} - \frac{1}{5} a - \frac{1}{2}$, $\frac{1}{10} a^{13} + \frac{1}{5} a^{10} - \frac{1}{10} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{3}{10} a^{6} + \frac{2}{5} a^{5} - \frac{1}{2} a^{4} - \frac{3}{10} a^{2} + \frac{3}{10} a$, $\frac{1}{20} a^{14} - \frac{1}{20} a^{11} + \frac{1}{20} a^{10} + \frac{1}{5} a^{9} - \frac{1}{2} a^{8} - \frac{3}{10} a^{7} + \frac{3}{10} a^{6} + \frac{1}{5} a^{5} - \frac{1}{4} a^{4} - \frac{3}{10} a^{3} - \frac{1}{2} a^{2} - \frac{1}{20} a - \frac{1}{4}$, $\frac{1}{20} a^{15} - \frac{1}{20} a^{12} - \frac{1}{20} a^{11} + \frac{1}{10} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{3}{10} a^{7} + \frac{1}{10} a^{6} - \frac{9}{20} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{7}{20} a^{2} - \frac{9}{20} a - \frac{1}{2}$, $\frac{1}{580} a^{16} + \frac{3}{580} a^{15} + \frac{1}{145} a^{14} - \frac{9}{580} a^{13} - \frac{6}{145} a^{12} + \frac{1}{580} a^{11} + \frac{13}{290} a^{10} - \frac{1}{290} a^{9} + \frac{9}{145} a^{8} - \frac{17}{290} a^{7} - \frac{259}{580} a^{6} - \frac{197}{580} a^{5} + \frac{13}{58} a^{4} + \frac{113}{580} a^{3} - \frac{19}{290} a^{2} + \frac{117}{580} a - \frac{5}{29}$, $\frac{1}{580} a^{17} - \frac{1}{116} a^{15} + \frac{2}{145} a^{14} + \frac{3}{580} a^{13} + \frac{3}{116} a^{12} - \frac{3}{290} a^{11} - \frac{109}{580} a^{10} - \frac{4}{145} a^{9} + \frac{9}{58} a^{8} + \frac{249}{580} a^{7} + \frac{1}{5} a^{6} + \frac{83}{580} a^{5} + \frac{5}{29} a^{4} - \frac{9}{20} a^{3} - \frac{117}{580} a^{2} - \frac{37}{290} a + \frac{31}{116}$, $\frac{1}{2900} a^{18} + \frac{1}{2900} a^{17} + \frac{1}{2900} a^{16} - \frac{2}{725} a^{15} + \frac{16}{725} a^{14} + \frac{11}{1450} a^{13} + \frac{63}{1450} a^{12} - \frac{109}{2900} a^{11} + \frac{7}{58} a^{10} - \frac{259}{1450} a^{9} + \frac{1251}{2900} a^{8} + \frac{1031}{2900} a^{7} + \frac{849}{2900} a^{6} + \frac{149}{725} a^{5} + \frac{59}{290} a^{4} - \frac{53}{1450} a^{3} - \frac{717}{1450} a^{2} - \frac{1}{4} a + \frac{23}{58}$, $\frac{1}{532846401215418197359400} a^{19} + \frac{160094818077501113}{1837401383501442059860} a^{18} - \frac{44043136337196045263}{53284640121541819735940} a^{17} + \frac{206925251423853315991}{532846401215418197359400} a^{16} + \frac{4952410289744288935987}{532846401215418197359400} a^{15} - \frac{7415136901205517863567}{532846401215418197359400} a^{14} - \frac{2801990408012459880549}{133211600303854549339850} a^{13} - \frac{2044014115879885191911}{53284640121541819735940} a^{12} + \frac{6250554336566035214889}{532846401215418197359400} a^{11} - \frac{4294970747668628086427}{18374013835014420598600} a^{10} - \frac{22475897507561732733461}{532846401215418197359400} a^{9} - \frac{97931671167415647831}{207333230044909804420} a^{8} - \frac{115425823000937329987251}{266423200607709098679700} a^{7} - \frac{3461189670154621555147}{18374013835014420598600} a^{6} - \frac{8804488683161345565831}{532846401215418197359400} a^{5} - \frac{203684035091044739084021}{532846401215418197359400} a^{4} + \frac{23931041674229821701514}{66605800151927274669925} a^{3} + \frac{73182627369887621726997}{266423200607709098679700} a^{2} - \frac{4602885826171004045941}{106569280243083639471880} a + \frac{3726555229819339947275}{21313856048616727894376}$

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:  \( -1 \) (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:  \( 4795512.92142 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{29}) \), 4.0.24389.1, 5.1.1975509.1 x5, 10.2.113176438463349.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.1975509.1
Degree 10 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
3Data not computed
$29$29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$