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Properties

Label 20.0.379...597.1
Degree $20$
Signature $[0, 10]$
Discriminant $3.794\times 10^{20}$
Root discriminant $10.69$
Ramified primes $3, 31$
Class number $1$
Class group trivial
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 6*x^18 + 14*x^17 - 46*x^16 + 25*x^15 + 73*x^14 - 122*x^13 - 8*x^12 + 193*x^11 - 146*x^10 - 144*x^9 + 308*x^8 - 127*x^7 - 128*x^6 + 161*x^5 - 46*x^4 - 26*x^3 + 25*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 6*x^18 + 14*x^17 - 46*x^16 + 25*x^15 + 73*x^14 - 122*x^13 - 8*x^12 + 193*x^11 - 146*x^10 - 144*x^9 + 308*x^8 - 127*x^7 - 128*x^6 + 161*x^5 - 46*x^4 - 26*x^3 + 25*x^2 - 8*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 25, -26, -46, 161, -128, -127, 308, -144, -146, 193, -8, -122, 73, 25, -46, 14, 6, -5, 1]);
 

\( x^{20} - 5 x^{19} + 6 x^{18} + 14 x^{17} - 46 x^{16} + 25 x^{15} + 73 x^{14} - 122 x^{13} - 8 x^{12} + 193 x^{11} - 146 x^{10} - 144 x^{9} + 308 x^{8} - 127 x^{7} - 128 x^{6} + 161 x^{5} - 46 x^{4} - 26 x^{3} + 25 x^{2} - 8 x + 1 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(379379679498607236597\)\(\medspace = 3^{15}\cdot 31^{9}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $10.69$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 31$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $10$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{6978527} a^{19} - \frac{502850}{6978527} a^{18} + \frac{2639465}{6978527} a^{17} - \frac{2706308}{6978527} a^{16} - \frac{1189948}{6978527} a^{15} - \frac{438476}{6978527} a^{14} - \frac{1096272}{6978527} a^{13} + \frac{110407}{6978527} a^{12} - \frac{3425638}{6978527} a^{11} - \frac{707323}{6978527} a^{10} - \frac{751820}{6978527} a^{9} + \frac{1184585}{6978527} a^{8} + \frac{3485122}{6978527} a^{7} - \frac{557869}{6978527} a^{6} - \frac{1191169}{6978527} a^{5} - \frac{574971}{6978527} a^{4} + \frac{918839}{6978527} a^{3} + \frac{718635}{6978527} a^{2} + \frac{68564}{6978527} a - \frac{3141208}{6978527}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{294846}{18413} a^{19} - \frac{1356362}{18413} a^{18} + \frac{1211607}{18413} a^{17} + \frac{4680751}{18413} a^{16} - \frac{11744765}{18413} a^{15} + \frac{2419699}{18413} a^{14} + \frac{23070963}{18413} a^{13} - \frac{26770987}{18413} a^{12} - \frac{14299518}{18413} a^{11} + \frac{52373757}{18413} a^{10} - \frac{21096924}{18413} a^{9} - \frac{53479978}{18413} a^{8} + \frac{69996438}{18413} a^{7} - \frac{6459642}{18413} a^{6} - \frac{43411115}{18413} a^{5} + \frac{29757726}{18413} a^{4} + \frac{526458}{18413} a^{3} - \frac{8417968}{18413} a^{2} + \frac{3705327}{18413} a - \frac{536855}{18413} \) (order $6$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 181.753135766 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{10}\cdot 181.753135766 \cdot 1}{6\sqrt{379379679498607236597}}\approx 0.149139373015$

Galois group

$C_5\times C_5:D_4$ (as 20T53):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.837.1, 10.0.224415603.1

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 $20$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
$31$31.10.9.3$x^{10} - 74431$$10$$1$$9$$C_{10}$$[\ ]_{10}$
31.10.0.1$x^{10} - x + 11$$1$$10$$0$$C_{10}$$[\ ]^{10}$