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Properties

Label 20.0.26483742361...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{16}\cdot 5^{15}\cdot 17^{10}$
Root discriminant $33.20$
Ramified primes $3, 5, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43025, 10150, 29985, -122455, 181936, -191825, 156405, -111880, 78876, -50875, 29250, -13170, 4691, -1590, 880, -640, 386, -165, 50, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 50*x^18 - 165*x^17 + 386*x^16 - 640*x^15 + 880*x^14 - 1590*x^13 + 4691*x^12 - 13170*x^11 + 29250*x^10 - 50875*x^9 + 78876*x^8 - 111880*x^7 + 156405*x^6 - 191825*x^5 + 181936*x^4 - 122455*x^3 + 29985*x^2 + 10150*x + 43025)
 
gp: K = bnfinit(x^20 - 10*x^19 + 50*x^18 - 165*x^17 + 386*x^16 - 640*x^15 + 880*x^14 - 1590*x^13 + 4691*x^12 - 13170*x^11 + 29250*x^10 - 50875*x^9 + 78876*x^8 - 111880*x^7 + 156405*x^6 - 191825*x^5 + 181936*x^4 - 122455*x^3 + 29985*x^2 + 10150*x + 43025, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 50 x^{18} - 165 x^{17} + 386 x^{16} - 640 x^{15} + 880 x^{14} - 1590 x^{13} + 4691 x^{12} - 13170 x^{11} + 29250 x^{10} - 50875 x^{9} + 78876 x^{8} - 111880 x^{7} + 156405 x^{6} - 191825 x^{5} + 181936 x^{4} - 122455 x^{3} + 29985 x^{2} + 10150 x + 43025 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2648374236155086600616455078125=3^{16}\cdot 5^{15}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.20$
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:  $3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{880} a^{16} - \frac{1}{110} a^{15} - \frac{9}{110} a^{14} + \frac{51}{220} a^{13} + \frac{11}{80} a^{12} + \frac{109}{440} a^{11} + \frac{3}{88} a^{10} + \frac{367}{880} a^{9} + \frac{1}{8} a^{8} + \frac{41}{110} a^{7} - \frac{409}{880} a^{6} - \frac{163}{440} a^{5} + \frac{37}{220} a^{4} - \frac{213}{880} a^{3} + \frac{23}{440} a^{2} - \frac{21}{176} a - \frac{75}{176}$, $\frac{1}{880} a^{17} - \frac{17}{110} a^{15} + \frac{17}{220} a^{14} - \frac{7}{880} a^{13} - \frac{67}{440} a^{12} - \frac{213}{440} a^{11} - \frac{273}{880} a^{10} - \frac{17}{440} a^{9} - \frac{7}{55} a^{8} - \frac{85}{176} a^{7} + \frac{181}{440} a^{6} - \frac{13}{44} a^{5} + \frac{91}{880} a^{4} - \frac{169}{440} a^{3} - \frac{177}{880} a^{2} - \frac{67}{176} a + \frac{1}{11}$, $\frac{1}{80211583745768998160} a^{18} - \frac{9}{80211583745768998160} a^{17} - \frac{32654317908304511}{80211583745768998160} a^{16} + \frac{65308635816609073}{20052895936442249540} a^{15} - \frac{4816993029741001339}{80211583745768998160} a^{14} - \frac{10958445171860125531}{80211583745768998160} a^{13} - \frac{100706626016275847}{943665691126694096} a^{12} + \frac{5177378856682672411}{80211583745768998160} a^{11} - \frac{10977965711776743147}{80211583745768998160} a^{10} - \frac{37557154079836360951}{80211583745768998160} a^{9} + \frac{1768407053893680903}{7291962158706272560} a^{8} - \frac{417110733274985567}{4221662302408894640} a^{7} - \frac{25225764712662459983}{80211583745768998160} a^{6} + \frac{24931931965920836561}{80211583745768998160} a^{5} - \frac{1513354749417524043}{4221662302408894640} a^{4} + \frac{29941948843752425}{235916422781673524} a^{3} + \frac{1128884199547010568}{5013223984110562385} a^{2} + \frac{1663665977511319719}{8021158374576899816} a - \frac{1950929934277789075}{16042316749153799632}$, $\frac{1}{470761785003918250201040} a^{19} + \frac{585}{94152357000783650040208} a^{18} + \frac{59763138740168395161}{117690446250979562550260} a^{17} - \frac{12771140226466701243}{23538089250195912510052} a^{16} + \frac{91607097318724338513637}{470761785003918250201040} a^{15} + \frac{39573317042001604699967}{470761785003918250201040} a^{14} - \frac{28192795855706551713097}{117690446250979562550260} a^{13} + \frac{2686110806849859134199}{42796525909447113654640} a^{12} + \frac{34206915357669364071977}{470761785003918250201040} a^{11} - \frac{3952406386729997738141}{21398262954723556827320} a^{10} + \frac{23712656612819141365719}{470761785003918250201040} a^{9} - \frac{185781853000933980150683}{470761785003918250201040} a^{8} + \frac{55817059565377603546013}{235380892501959125100520} a^{7} - \frac{220115466610595956346937}{470761785003918250201040} a^{6} - \frac{22316335848721894109975}{94152357000783650040208} a^{5} + \frac{6666892206615935013889}{470761785003918250201040} a^{4} - \frac{55733440290765640154139}{117690446250979562550260} a^{3} - \frac{155520455562657733091183}{470761785003918250201040} a^{2} + \frac{3892965354502523516779}{23538089250195912510052} a + \frac{2476622520700065593893}{5884522312548978127513}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 8803149.53857 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.36125.1, 5.1.2926125.1 x5, 10.2.42811037578125.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.2926125.1
Degree 10 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
3Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$17$17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$