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Properties

Label 20.20.2448962526...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{16}\cdot 5^{10}\cdot 13^{4}\cdot 29^{9}\cdot 31^{4}$
Root discriminant $58.81$
Ramified primes $2, 5, 13, 29, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T658

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, 86, -2100, 3143, 18689, -63952, 45466, 70104, -110376, -4842, 78290, -22722, -24607, 11758, 3322, -2417, -78, 218, -18, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 - 18*x^18 + 218*x^17 - 78*x^16 - 2417*x^15 + 3322*x^14 + 11758*x^13 - 24607*x^12 - 22722*x^11 + 78290*x^10 - 4842*x^9 - 110376*x^8 + 70104*x^7 + 45466*x^6 - 63952*x^5 + 18689*x^4 + 3143*x^3 - 2100*x^2 + 86*x + 41)
 
gp: K = bnfinit(x^20 - 7*x^19 - 18*x^18 + 218*x^17 - 78*x^16 - 2417*x^15 + 3322*x^14 + 11758*x^13 - 24607*x^12 - 22722*x^11 + 78290*x^10 - 4842*x^9 - 110376*x^8 + 70104*x^7 + 45466*x^6 - 63952*x^5 + 18689*x^4 + 3143*x^3 - 2100*x^2 + 86*x + 41, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} - 18 x^{18} + 218 x^{17} - 78 x^{16} - 2417 x^{15} + 3322 x^{14} + 11758 x^{13} - 24607 x^{12} - 22722 x^{11} + 78290 x^{10} - 4842 x^{9} - 110376 x^{8} + 70104 x^{7} + 45466 x^{6} - 63952 x^{5} + 18689 x^{4} + 3143 x^{3} - 2100 x^{2} + 86 x + 41 \)

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:  \(244896252618707380317560960000000000=2^{16}\cdot 5^{10}\cdot 13^{4}\cdot 29^{9}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.81$
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:  $2, 5, 13, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{26} a^{18} - \frac{3}{13} a^{17} + \frac{5}{26} a^{16} + \frac{5}{13} a^{15} - \frac{1}{26} a^{14} + \frac{2}{13} a^{13} - \frac{5}{26} a^{12} + \frac{1}{13} a^{9} + \frac{3}{13} a^{8} + \frac{3}{13} a^{7} - \frac{4}{13} a^{6} - \frac{4}{13} a^{5} + \frac{6}{13} a^{4} - \frac{2}{13} a^{3} + \frac{1}{26} a^{2} + \frac{6}{13} a - \frac{5}{26}$, $\frac{1}{9730794486368695780924546} a^{19} + \frac{66575623406262828230482}{4865397243184347890462273} a^{18} - \frac{29103803307565028563829}{374261326398795991574021} a^{17} - \frac{2202341811933300393304859}{9730794486368695780924546} a^{16} + \frac{2257062573907654850017047}{9730794486368695780924546} a^{15} + \frac{2019226261801650305443670}{4865397243184347890462273} a^{14} + \frac{252215033161859356006636}{4865397243184347890462273} a^{13} + \frac{3049524130805125195219775}{9730794486368695780924546} a^{12} - \frac{144394350291452817538528}{374261326398795991574021} a^{11} + \frac{252121377147966536542996}{4865397243184347890462273} a^{10} - \frac{685506404015663800909515}{4865397243184347890462273} a^{9} - \frac{453476828018415115384055}{4865397243184347890462273} a^{8} + \frac{279352104073429713082320}{4865397243184347890462273} a^{7} + \frac{1755768364235453729157214}{4865397243184347890462273} a^{6} - \frac{703394481136494267466469}{4865397243184347890462273} a^{5} - \frac{2287919157968506709507494}{4865397243184347890462273} a^{4} - \frac{1347635532792742764968967}{9730794486368695780924546} a^{3} - \frac{116473284620657945370649}{4865397243184347890462273} a^{2} + \frac{458517152431075483895028}{4865397243184347890462273} a - \frac{1809230478647864155035843}{9730794486368695780924546}$

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:  \( 171196137065 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T658:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 76 conjugacy class representatives for t20n658 are not computed
Character table for t20n658 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 10.10.109268775200000.1

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 24 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 $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
29.10.5.2$x^{10} - 707281 x^{2} + 225622639$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$