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Properties

Label 20.0.45037507812...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{23}\cdot 7^{8}$
Root discriminant $24.14$
Ramified primes $2, 5, 7$
Class number $1$ (GRH)
Class group Trivial (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![311, -1305, 1630, -1800, 3950, -2261, 490, -2190, 335, 470, 356, 290, -30, -100, -60, -46, 15, -5, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 10*x^18 - 5*x^17 + 15*x^16 - 46*x^15 - 60*x^14 - 100*x^13 - 30*x^12 + 290*x^11 + 356*x^10 + 470*x^9 + 335*x^8 - 2190*x^7 + 490*x^6 - 2261*x^5 + 3950*x^4 - 1800*x^3 + 1630*x^2 - 1305*x + 311)
 
gp: K = bnfinit(x^20 + 10*x^18 - 5*x^17 + 15*x^16 - 46*x^15 - 60*x^14 - 100*x^13 - 30*x^12 + 290*x^11 + 356*x^10 + 470*x^9 + 335*x^8 - 2190*x^7 + 490*x^6 - 2261*x^5 + 3950*x^4 - 1800*x^3 + 1630*x^2 - 1305*x + 311, 1)
 

Normalized defining polynomial

\( x^{20} + 10 x^{18} - 5 x^{17} + 15 x^{16} - 46 x^{15} - 60 x^{14} - 100 x^{13} - 30 x^{12} + 290 x^{11} + 356 x^{10} + 470 x^{9} + 335 x^{8} - 2190 x^{7} + 490 x^{6} - 2261 x^{5} + 3950 x^{4} - 1800 x^{3} + 1630 x^{2} - 1305 x + 311 \)

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:  \(4503750781250000000000000000=2^{16}\cdot 5^{23}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.14$
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 a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{41} a^{17} + \frac{1}{41} a^{16} - \frac{8}{41} a^{15} + \frac{10}{41} a^{14} + \frac{14}{41} a^{13} + \frac{16}{41} a^{12} - \frac{13}{41} a^{11} + \frac{7}{41} a^{10} - \frac{6}{41} a^{9} + \frac{15}{41} a^{8} + \frac{15}{41} a^{6} - \frac{4}{41} a^{5} - \frac{19}{41} a^{4} - \frac{12}{41} a^{3} + \frac{11}{41} a^{2} - \frac{12}{41} a + \frac{17}{41}$, $\frac{1}{200882821} a^{18} + \frac{23394}{200882821} a^{17} - \frac{29371934}{200882821} a^{16} - \frac{68230980}{200882821} a^{15} + \frac{26531959}{200882821} a^{14} + \frac{29308983}{200882821} a^{13} - \frac{2679939}{6480091} a^{12} + \frac{17711380}{200882821} a^{11} + \frac{24907409}{200882821} a^{10} + \frac{894692}{4899581} a^{9} - \frac{23504053}{200882821} a^{8} + \frac{1853748}{200882821} a^{7} - \frac{579032}{6480091} a^{6} - \frac{57058827}{200882821} a^{5} - \frac{20156623}{200882821} a^{4} - \frac{76927745}{200882821} a^{3} + \frac{29208682}{200882821} a^{2} - \frac{5415088}{200882821} a - \frac{7372762}{200882821}$, $\frac{1}{265211441365411257188741} a^{19} + \frac{293389427547118}{265211441365411257188741} a^{18} - \frac{2071913203688286125800}{265211441365411257188741} a^{17} - \frac{2627423194445934871633}{265211441365411257188741} a^{16} - \frac{109055450576072439636547}{265211441365411257188741} a^{15} - \frac{17153702307236836390548}{265211441365411257188741} a^{14} + \frac{59516845134449300158915}{265211441365411257188741} a^{13} + \frac{91238792102745078812798}{265211441365411257188741} a^{12} + \frac{70998739796281297729794}{265211441365411257188741} a^{11} - \frac{19776578619456094043337}{265211441365411257188741} a^{10} - \frac{70121041211402415246332}{265211441365411257188741} a^{9} - \frac{37467176382982158351025}{265211441365411257188741} a^{8} - \frac{91853445711339048620010}{265211441365411257188741} a^{7} + \frac{1443924939450411641850}{6468571740619786760701} a^{6} - \frac{96934559735992227482218}{265211441365411257188741} a^{5} + \frac{70896964647446206333219}{265211441365411257188741} a^{4} - \frac{68834216518026778772324}{265211441365411257188741} a^{3} + \frac{89294445071102298366309}{265211441365411257188741} a^{2} + \frac{44370924722007255008401}{265211441365411257188741} a + \frac{112286064950297335046282}{265211441365411257188741}$

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:  \( -\frac{3233097247159695276582}{265211441365411257188741} a^{19} - \frac{2304872176461308450505}{265211441365411257188741} a^{18} - \frac{34060215020755136023242}{265211441365411257188741} a^{17} - \frac{8563414051689290252386}{265211441365411257188741} a^{16} - \frac{56126025910608477111698}{265211441365411257188741} a^{15} + \frac{104088837467796157961754}{265211441365411257188741} a^{14} + \frac{261661011523644777129362}{265211441365411257188741} a^{13} + \frac{503484675895797895073545}{265211441365411257188741} a^{12} + \frac{14995990564484536983060}{8555207785981008296411} a^{11} - \frac{557006576645865517102498}{265211441365411257188741} a^{10} - \frac{1454350666734973446778788}{265211441365411257188741} a^{9} - \frac{2450204533436723636962949}{265211441365411257188741} a^{8} - \frac{2828585173379288585840286}{265211441365411257188741} a^{7} + \frac{4861303386658572703499395}{265211441365411257188741} a^{6} + \frac{1476702935198979081116938}{265211441365411257188741} a^{5} + \frac{7852841578706656396278201}{265211441365411257188741} a^{4} - \frac{7106395083703816379212406}{265211441365411257188741} a^{3} + \frac{1163577161281567567034627}{265211441365411257188741} a^{2} - \frac{3924035768438624331209896}{265211441365411257188741} a + \frac{1743746937050292597133587}{265211441365411257188741} \) (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:  \( 1633832.53547 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.5.2450000.1, 10.10.30012500000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
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}$