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Properties

Label 20.10.2584296997...6848.2
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{38}\cdot 7^{9}\cdot 13^{12}$
Root discriminant $41.75$
Ramified primes $2, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T633

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![350, 160, -1603, 1754, 5377, -2860, -9441, 782, 10889, 3332, -5832, -3036, 1070, 676, -164, -60, 74, 16, -15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 15*x^18 + 16*x^17 + 74*x^16 - 60*x^15 - 164*x^14 + 676*x^13 + 1070*x^12 - 3036*x^11 - 5832*x^10 + 3332*x^9 + 10889*x^8 + 782*x^7 - 9441*x^6 - 2860*x^5 + 5377*x^4 + 1754*x^3 - 1603*x^2 + 160*x + 350)
 
gp: K = bnfinit(x^20 - 2*x^19 - 15*x^18 + 16*x^17 + 74*x^16 - 60*x^15 - 164*x^14 + 676*x^13 + 1070*x^12 - 3036*x^11 - 5832*x^10 + 3332*x^9 + 10889*x^8 + 782*x^7 - 9441*x^6 - 2860*x^5 + 5377*x^4 + 1754*x^3 - 1603*x^2 + 160*x + 350, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 15 x^{18} + 16 x^{17} + 74 x^{16} - 60 x^{15} - 164 x^{14} + 676 x^{13} + 1070 x^{12} - 3036 x^{11} - 5832 x^{10} + 3332 x^{9} + 10889 x^{8} + 782 x^{7} - 9441 x^{6} - 2860 x^{5} + 5377 x^{4} + 1754 x^{3} - 1603 x^{2} + 160 x + 350 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-258429699769673174020904974286848=-\,2^{38}\cdot 7^{9}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.75$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{5} a^{17} - \frac{2}{5} a^{16} + \frac{2}{5} a^{15} + \frac{2}{5} a^{14} - \frac{2}{5} a^{13} - \frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{35} a^{18} + \frac{2}{35} a^{17} - \frac{6}{35} a^{16} + \frac{3}{7} a^{15} + \frac{16}{35} a^{14} - \frac{9}{35} a^{13} - \frac{2}{35} a^{12} - \frac{13}{35} a^{11} + \frac{3}{7} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{7} a^{6} - \frac{1}{5} a^{5} - \frac{8}{35} a^{4} - \frac{8}{35} a^{3} - \frac{4}{35} a^{2} + \frac{1}{35} a$, $\frac{1}{3762236386345275064545088370191010} a^{19} - \frac{46897811021835982336489808367421}{3762236386345275064545088370191010} a^{18} - \frac{162189591494480278362178327805759}{1881118193172637532272544185095505} a^{17} - \frac{158889151720653633565804552480058}{376223638634527506454508837019101} a^{16} - \frac{288501186772128530809451118026438}{1881118193172637532272544185095505} a^{15} - \frac{107638945826809325175274020463871}{268731170453233933181792026442215} a^{14} - \frac{916060471283652386306955235569214}{1881118193172637532272544185095505} a^{13} + \frac{376503032804021958506966864417312}{1881118193172637532272544185095505} a^{12} + \frac{764138639353003293560142629406096}{1881118193172637532272544185095505} a^{11} + \frac{243078772652718723517244688098842}{1881118193172637532272544185095505} a^{10} + \frac{92427019631749686091644217894054}{268731170453233933181792026442215} a^{9} - \frac{40600795242786827266749894882326}{268731170453233933181792026442215} a^{8} + \frac{1442879882151456683830947263289879}{3762236386345275064545088370191010} a^{7} - \frac{527919063508546813517818578591483}{3762236386345275064545088370191010} a^{6} - \frac{322947727572885660593983702373989}{1881118193172637532272544185095505} a^{5} + \frac{546904365014555336935487290456716}{1881118193172637532272544185095505} a^{4} - \frac{1820786376405945659159796383515981}{3762236386345275064545088370191010} a^{3} - \frac{319242193955622260945307967124559}{752447277269055012909017674038202} a^{2} - \frac{833380731382676803368670934776746}{1881118193172637532272544185095505} a - \frac{15022604325777359650137939623140}{53746234090646786636358405288443}$

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:  $14$
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:  \( 4302704642.32 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T633:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40960
The 124 conjugacy class representatives for t20n633 are not computed
Character table for t20n633 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.5.6889792.1, 10.10.379753870426112.1

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 R $20$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
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}$
13Data not computed