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Properties

Label 20.0.15152383971...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 83^{10}$
Root discriminant $20.37$
Ramified primes $5, 83$
Class number $3$
Class group $[3]$
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![441, -168, -256, -347, 680, 17, -1284, 956, 449, -849, 505, -247, 401, -294, 85, -31, 30, -7, 1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + x^18 - 7*x^17 + 30*x^16 - 31*x^15 + 85*x^14 - 294*x^13 + 401*x^12 - 247*x^11 + 505*x^10 - 849*x^9 + 449*x^8 + 956*x^7 - 1284*x^6 + 17*x^5 + 680*x^4 - 347*x^3 - 256*x^2 - 168*x + 441)
 
gp: K = bnfinit(x^20 - 2*x^19 + x^18 - 7*x^17 + 30*x^16 - 31*x^15 + 85*x^14 - 294*x^13 + 401*x^12 - 247*x^11 + 505*x^10 - 849*x^9 + 449*x^8 + 956*x^7 - 1284*x^6 + 17*x^5 + 680*x^4 - 347*x^3 - 256*x^2 - 168*x + 441, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + x^{18} - 7 x^{17} + 30 x^{16} - 31 x^{15} + 85 x^{14} - 294 x^{13} + 401 x^{12} - 247 x^{11} + 505 x^{10} - 849 x^{9} + 449 x^{8} + 956 x^{7} - 1284 x^{6} + 17 x^{5} + 680 x^{4} - 347 x^{3} - 256 x^{2} - 168 x + 441 \)

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:  \(151523839718807162587890625=5^{10}\cdot 83^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.37$
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:  $5, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{12} - \frac{1}{9} a^{8} - \frac{2}{9} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{189} a^{15} + \frac{1}{27} a^{14} - \frac{20}{189} a^{13} + \frac{4}{189} a^{12} - \frac{4}{63} a^{11} + \frac{8}{63} a^{10} - \frac{4}{27} a^{9} + \frac{2}{27} a^{8} - \frac{5}{63} a^{7} + \frac{2}{9} a^{6} - \frac{83}{189} a^{5} - \frac{44}{189} a^{4} - \frac{2}{189} a^{3} - \frac{47}{189} a^{2} - \frac{13}{63} a - \frac{1}{3}$, $\frac{1}{189} a^{16} - \frac{2}{63} a^{14} + \frac{2}{21} a^{13} + \frac{23}{189} a^{12} - \frac{2}{21} a^{11} - \frac{1}{27} a^{10} + \frac{1}{9} a^{9} + \frac{13}{189} a^{8} + \frac{1}{9} a^{7} + \frac{1}{189} a^{6} + \frac{11}{63} a^{5} - \frac{1}{21} a^{4} - \frac{11}{63} a^{3} - \frac{25}{189} a^{2} - \frac{2}{9} a + \frac{1}{3}$, $\frac{1}{189} a^{17} - \frac{1}{63} a^{14} + \frac{29}{189} a^{13} + \frac{2}{63} a^{12} - \frac{16}{189} a^{11} - \frac{8}{63} a^{10} - \frac{29}{189} a^{9} - \frac{1}{9} a^{8} - \frac{26}{189} a^{7} - \frac{31}{63} a^{6} + \frac{20}{63} a^{5} - \frac{5}{21} a^{4} - \frac{37}{189} a^{3} - \frac{1}{21} a^{2} + \frac{2}{21} a$, $\frac{1}{17955} a^{18} - \frac{2}{2565} a^{17} + \frac{13}{17955} a^{16} + \frac{26}{17955} a^{15} - \frac{28}{513} a^{14} + \frac{73}{17955} a^{13} - \frac{41}{855} a^{12} - \frac{593}{3591} a^{11} + \frac{16}{315} a^{10} - \frac{13}{2565} a^{9} + \frac{185}{1197} a^{8} - \frac{2663}{17955} a^{7} + \frac{632}{3591} a^{6} - \frac{1183}{2565} a^{5} - \frac{527}{17955} a^{4} + \frac{3802}{17955} a^{3} - \frac{1397}{17955} a^{2} - \frac{326}{1197} a + \frac{92}{285}$, $\frac{1}{486545947051939046685} a^{19} - \frac{13223880359030353}{486545947051939046685} a^{18} - \frac{14154553098748966}{23168854621520906985} a^{17} - \frac{1009585132814495221}{486545947051939046685} a^{16} + \frac{34150251026916763}{28620349826584649805} a^{15} + \frac{24042893091097854448}{486545947051939046685} a^{14} + \frac{55582770939735195392}{486545947051939046685} a^{13} - \frac{2661365824732852568}{28620349826584649805} a^{12} - \frac{1375349425675804529}{69506563864562720955} a^{11} - \frac{613459063801682029}{23168854621520906985} a^{10} + \frac{12402103570635132559}{486545947051939046685} a^{9} + \frac{71413083073623500647}{486545947051939046685} a^{8} - \frac{8474127588088223006}{162181982350646348895} a^{7} + \frac{130448784778629664049}{486545947051939046685} a^{6} + \frac{12411119331288379232}{486545947051939046685} a^{5} + \frac{44354182811290162409}{97309189410387809337} a^{4} - \frac{12499116523396882487}{32436396470129269779} a^{3} + \frac{61140042725397846883}{486545947051939046685} a^{2} + \frac{783270152881758781}{9540116608861549935} a - \frac{2929034176821979798}{7722951540506968995}$

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

Class group and class number

$C_{3}$, which has order $3$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 44240.6265271 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-415}) \), \(\Q(\sqrt{-83}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-83})\), 5.1.172225.1 x5, 10.0.12309502009375.1, 10.0.2461900401875.1 x5, 10.2.148307253125.1 x5

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

Sibling fields

Degree 10 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$83$83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$