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Properties

Label 20.12.6125762557...0496.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{48}\cdot 31^{10}\cdot 227^{4}$
Root discriminant $86.97$
Ramified primes $2, 31, 227$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![63032, 533536, -46599072, 87451344, -65906436, 51004208, -33430872, 567192, 9733290, -4641000, 572560, 626076, -296034, -20368, 27942, -1944, -944, 244, -10, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 - 10*x^18 + 244*x^17 - 944*x^16 - 1944*x^15 + 27942*x^14 - 20368*x^13 - 296034*x^12 + 626076*x^11 + 572560*x^10 - 4641000*x^9 + 9733290*x^8 + 567192*x^7 - 33430872*x^6 + 51004208*x^5 - 65906436*x^4 + 87451344*x^3 - 46599072*x^2 + 533536*x + 63032)
 
gp: K = bnfinit(x^20 - 8*x^19 - 10*x^18 + 244*x^17 - 944*x^16 - 1944*x^15 + 27942*x^14 - 20368*x^13 - 296034*x^12 + 626076*x^11 + 572560*x^10 - 4641000*x^9 + 9733290*x^8 + 567192*x^7 - 33430872*x^6 + 51004208*x^5 - 65906436*x^4 + 87451344*x^3 - 46599072*x^2 + 533536*x + 63032, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} - 10 x^{18} + 244 x^{17} - 944 x^{16} - 1944 x^{15} + 27942 x^{14} - 20368 x^{13} - 296034 x^{12} + 626076 x^{11} + 572560 x^{10} - 4641000 x^{9} + 9733290 x^{8} + 567192 x^{7} - 33430872 x^{6} + 51004208 x^{5} - 65906436 x^{4} + 87451344 x^{3} - 46599072 x^{2} + 533536 x + 63032 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(612576255759573370734751264818904170496=2^{48}\cdot 31^{10}\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.97$
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:  $2, 31, 227$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{17} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{18} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{123744470803851680485138522094642580994935144211737267589904978837862757827492} a^{19} - \frac{3176691081633753413394237605347499137148472839544873119763926928116746805967}{41248156934617226828379507364880860331645048070579089196634992945954252609164} a^{18} - \frac{3282766185232550459938876321602277574916484579245325990135522675444017567454}{30936117700962920121284630523660645248733786052934316897476244709465689456873} a^{17} + \frac{1911385476648328706412026904925216961774323441279867366599094567680589535041}{61872235401925840242569261047321290497467572105868633794952489418931378913746} a^{16} - \frac{6227450646183356078831050210689373901907393152234471954361386600692609322411}{61872235401925840242569261047321290497467572105868633794952489418931378913746} a^{15} - \frac{6584609456471276372813611382629467066037486367092446368416792728799202635739}{30936117700962920121284630523660645248733786052934316897476244709465689456873} a^{14} - \frac{10017776014013090661181068866070773555518169730188509585372390220399192579597}{61872235401925840242569261047321290497467572105868633794952489418931378913746} a^{13} + \frac{286498047703909167741839178192518062400703168580228301745098884266619677078}{10312039233654306707094876841220215082911262017644772299158748236488563152291} a^{12} - \frac{7684662556820626908154623219722174206588971549412952871099779757709506519625}{20624078467308613414189753682440430165822524035289544598317496472977126304582} a^{11} - \frac{6949488214031443067388136805147310048778550830767692026440966330495422698617}{20624078467308613414189753682440430165822524035289544598317496472977126304582} a^{10} + \frac{7237259565196739361072796059811903565363969597775443907797301366136348412968}{30936117700962920121284630523660645248733786052934316897476244709465689456873} a^{9} - \frac{2677388712112876420629884659896682807145209746744102103478174561009245350430}{30936117700962920121284630523660645248733786052934316897476244709465689456873} a^{8} + \frac{22021782986289358443421820787964030443647146017809877187969129339378891516427}{61872235401925840242569261047321290497467572105868633794952489418931378913746} a^{7} - \frac{15208464453815354518544425598342783918830038639032216411761516788127533369713}{61872235401925840242569261047321290497467572105868633794952489418931378913746} a^{6} - \frac{9905896507060382840292822117903403387304780664456917305888641004727527585808}{30936117700962920121284630523660645248733786052934316897476244709465689456873} a^{5} - \frac{468110290396331893605799791192054265616163932209821626894640202176203353859}{30936117700962920121284630523660645248733786052934316897476244709465689456873} a^{4} + \frac{1985724660053452295606395850447571715859205579864430381220728172003005866737}{30936117700962920121284630523660645248733786052934316897476244709465689456873} a^{3} + \frac{15295700018324850301694842407199280385140491712949462171431963103110910246907}{30936117700962920121284630523660645248733786052934316897476244709465689456873} a^{2} + \frac{14844707937936497654363912767020014290912418411715460451014369981864522734632}{30936117700962920121284630523660645248733786052934316897476244709465689456873} a - \frac{10914374917485156256024086189829133225944404114414002572335996762153897794053}{30936117700962920121284630523660645248733786052934316897476244709465689456873}$

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:  $15$
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:  \( 4873811636050 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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

Sibling fields

Degree 20 sibling: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ $16{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ $16{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.12.26.27$x^{12} - 2 x^{10} + 4 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{3} + 2$$12$$1$$26$12T48$[4/3, 4/3, 2, 3]_{3}^{2}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$
31.10.5.1$x^{10} - 1922 x^{6} + 923521 x^{2} - 2862915100$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
227Data not computed