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Properties

Label 20.4.23930921644...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 3^{10}\cdot 5^{22}\cdot 11^{10}$
Root discriminant $58.74$
Ramified primes $2, 3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3444019951, 97709600, -3271307835, -22732170, 1428543135, -1148286, -382460220, 899850, 69990105, -125530, -9154543, 8230, 864535, -190, -58060, -4, 2655, 0, -75, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 75*x^18 + 2655*x^16 - 4*x^15 - 58060*x^14 - 190*x^13 + 864535*x^12 + 8230*x^11 - 9154543*x^10 - 125530*x^9 + 69990105*x^8 + 899850*x^7 - 382460220*x^6 - 1148286*x^5 + 1428543135*x^4 - 22732170*x^3 - 3271307835*x^2 + 97709600*x + 3444019951)
 
gp: K = bnfinit(x^20 - 75*x^18 + 2655*x^16 - 4*x^15 - 58060*x^14 - 190*x^13 + 864535*x^12 + 8230*x^11 - 9154543*x^10 - 125530*x^9 + 69990105*x^8 + 899850*x^7 - 382460220*x^6 - 1148286*x^5 + 1428543135*x^4 - 22732170*x^3 - 3271307835*x^2 + 97709600*x + 3444019951, 1)
 

Normalized defining polynomial

\( x^{20} - 75 x^{18} + 2655 x^{16} - 4 x^{15} - 58060 x^{14} - 190 x^{13} + 864535 x^{12} + 8230 x^{11} - 9154543 x^{10} - 125530 x^{9} + 69990105 x^{8} + 899850 x^{7} - 382460220 x^{6} - 1148286 x^{5} + 1428543135 x^{4} - 22732170 x^{3} - 3271307835 x^{2} + 97709600 x + 3444019951 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(239309216447570156250000000000000000=2^{16}\cdot 3^{10}\cdot 5^{22}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.74$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{14} a^{18} - \frac{3}{14} a^{17} + \frac{3}{14} a^{16} + \frac{1}{14} a^{15} + \frac{3}{14} a^{14} + \frac{1}{7} a^{12} - \frac{1}{7} a^{10} + \frac{2}{7} a^{9} + \frac{5}{14} a^{8} - \frac{2}{7} a^{7} + \frac{3}{14} a^{6} - \frac{5}{14} a^{5} - \frac{2}{7} a^{4} - \frac{3}{14} a^{3} - \frac{3}{14} a^{2} + \frac{3}{7} a + \frac{1}{14}$, $\frac{1}{25882932252342560970679673304349870245092618246943953030262986936798} a^{19} - \frac{49657083968194570836837401264530583828916901182218560847481426223}{3697561750334651567239953329192838606441802606706279004323283848114} a^{18} + \frac{3060663325360665260603385284856772279188710684463277184396818459212}{12941466126171280485339836652174935122546309123471976515131493468399} a^{17} - \frac{1675730194517964173286162054445513107582302131167627782552230016525}{12941466126171280485339836652174935122546309123471976515131493468399} a^{16} + \frac{255330328834557708922868981249643293088877001148083663963853221069}{12941466126171280485339836652174935122546309123471976515131493468399} a^{15} - \frac{23307450347126080059613199591589620040188836847753084291525112533}{681129796114277920281044034324996585397174164393261921849025972021} a^{14} - \frac{1328715129111996828249078329873993712793340494354884432438449896228}{12941466126171280485339836652174935122546309123471976515131493468399} a^{13} + \frac{1764323148611956311858307941024100298021720172163447497152732246989}{12941466126171280485339836652174935122546309123471976515131493468399} a^{12} + \frac{2545810025209901473613973393734814355880542034548765778126726396664}{12941466126171280485339836652174935122546309123471976515131493468399} a^{11} - \frac{155795469690497825436999047125900542393992130897793569003900016624}{681129796114277920281044034324996585397174164393261921849025972021} a^{10} + \frac{4492458227718190474573515581442045659243335051530627740531980281355}{25882932252342560970679673304349870245092618246943953030262986936798} a^{9} + \frac{1243720378559729298980432284032876242105995654922127370161205401612}{12941466126171280485339836652174935122546309123471976515131493468399} a^{8} - \frac{5205847161412056140923966178842608270881321898083458845054289351595}{25882932252342560970679673304349870245092618246943953030262986936798} a^{7} + \frac{7005447552414964566585481041907859161873603778896986864671499184293}{25882932252342560970679673304349870245092618246943953030262986936798} a^{6} + \frac{9868490698642719100903170088544576244808062897656126822587978188567}{25882932252342560970679673304349870245092618246943953030262986936798} a^{5} + \frac{6343300364731231799615950605368475061432163009560648422667217591763}{12941466126171280485339836652174935122546309123471976515131493468399} a^{4} + \frac{6100759535284120068590861079571108250652373396949267538992457440788}{12941466126171280485339836652174935122546309123471976515131493468399} a^{3} + \frac{744115955410659479699872500738434701274634370816018745213782951948}{12941466126171280485339836652174935122546309123471976515131493468399} a^{2} + \frac{2991454570791683692270412520099531832359694121543005789420415726507}{25882932252342560970679673304349870245092618246943953030262986936798} a - \frac{4174726979641987600542200294223126662211781667193483958248716681362}{12941466126171280485339836652174935122546309123471976515131493468399}$

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:  $11$
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:  \( 6685106585.539339 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{165}) \), \(\Q(\sqrt{5}, \sqrt{33})\), 5.1.50000.1, 10.2.12500000000.1, 10.2.97838482500000000.1, 10.2.489192412500000000.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
$11$11.10.5.1$x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.5.1$x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$