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Properties

Label 20.20.2019979533...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{18}\cdot 5^{10}\cdot 307^{8}$
Root discriminant $41.24$
Ramified primes $2, 5, 307$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times S_5$ (as 20T62)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5431, -9313, -55470, 103459, 105445, -295673, 15158, 309409, -143822, -137699, 114437, 18927, -39284, 4692, 6363, -1897, -387, 219, -7, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 - 7*x^18 + 219*x^17 - 387*x^16 - 1897*x^15 + 6363*x^14 + 4692*x^13 - 39284*x^12 + 18927*x^11 + 114437*x^10 - 137699*x^9 - 143822*x^8 + 309409*x^7 + 15158*x^6 - 295673*x^5 + 105445*x^4 + 103459*x^3 - 55470*x^2 - 9313*x + 5431)
 
gp: K = bnfinit(x^20 - 8*x^19 - 7*x^18 + 219*x^17 - 387*x^16 - 1897*x^15 + 6363*x^14 + 4692*x^13 - 39284*x^12 + 18927*x^11 + 114437*x^10 - 137699*x^9 - 143822*x^8 + 309409*x^7 + 15158*x^6 - 295673*x^5 + 105445*x^4 + 103459*x^3 - 55470*x^2 - 9313*x + 5431, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} - 7 x^{18} + 219 x^{17} - 387 x^{16} - 1897 x^{15} + 6363 x^{14} + 4692 x^{13} - 39284 x^{12} + 18927 x^{11} + 114437 x^{10} - 137699 x^{9} - 143822 x^{8} + 309409 x^{7} + 15158 x^{6} - 295673 x^{5} + 105445 x^{4} + 103459 x^{3} - 55470 x^{2} - 9313 x + 5431 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(201997953325162874882560000000000=2^{18}\cdot 5^{10}\cdot 307^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.24$
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:  $2, 5, 307$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11} a^{18} - \frac{2}{11} a^{17} + \frac{4}{11} a^{16} - \frac{1}{11} a^{15} - \frac{4}{11} a^{14} + \frac{3}{11} a^{13} - \frac{3}{11} a^{12} + \frac{2}{11} a^{11} - \frac{5}{11} a^{10} + \frac{1}{11} a^{9} + \frac{5}{11} a^{8} - \frac{3}{11} a^{7} + \frac{1}{11} a^{6} + \frac{4}{11} a^{5} + \frac{3}{11} a^{4} - \frac{4}{11} a^{3} + \frac{3}{11}$, $\frac{1}{110682730818169807} a^{19} + \frac{562215352932296}{110682730818169807} a^{18} - \frac{51543908954166601}{110682730818169807} a^{17} - \frac{33615920853595503}{110682730818169807} a^{16} + \frac{2863885732046666}{110682730818169807} a^{15} - \frac{26736621329228463}{110682730818169807} a^{14} - \frac{846164966367969}{110682730818169807} a^{13} - \frac{13597328584234676}{110682730818169807} a^{12} - \frac{36883028776913546}{110682730818169807} a^{11} - \frac{52863620930725478}{110682730818169807} a^{10} - \frac{44532501218277365}{110682730818169807} a^{9} + \frac{23985591106739672}{110682730818169807} a^{8} + \frac{38579322410586709}{110682730818169807} a^{7} - \frac{52719801845076605}{110682730818169807} a^{6} - \frac{43661442372641996}{110682730818169807} a^{5} - \frac{40205618326342971}{110682730818169807} a^{4} + \frac{457599318339013}{110682730818169807} a^{3} - \frac{2392623095207295}{10062066438015437} a^{2} + \frac{24155960859790884}{110682730818169807} a - \frac{8611985978957078}{110682730818169807}$

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:  $19$
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:  \( 6896317800.91 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_5$ (as 20T62):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 240
The 14 conjugacy class representatives for $C_2\times S_5$
Character table for $C_2\times S_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.568503936064000.1

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
Degree 12 siblings: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 30 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
307Data not computed