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Properties

Label 20.0.30631579705...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{10}\cdot 5^{15}\cdot 11^{10}$
Root discriminant $33.44$
Ramified primes $2, 3, 5, 11$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (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![96595, -163125, 311630, -389860, 468466, -458653, 417836, -336594, 256621, -179424, 118256, -71194, 39326, -19384, 8514, -3248, 1069, -291, 64, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 64*x^18 - 291*x^17 + 1069*x^16 - 3248*x^15 + 8514*x^14 - 19384*x^13 + 39326*x^12 - 71194*x^11 + 118256*x^10 - 179424*x^9 + 256621*x^8 - 336594*x^7 + 417836*x^6 - 458653*x^5 + 468466*x^4 - 389860*x^3 + 311630*x^2 - 163125*x + 96595)
 
gp: K = bnfinit(x^20 - 10*x^19 + 64*x^18 - 291*x^17 + 1069*x^16 - 3248*x^15 + 8514*x^14 - 19384*x^13 + 39326*x^12 - 71194*x^11 + 118256*x^10 - 179424*x^9 + 256621*x^8 - 336594*x^7 + 417836*x^6 - 458653*x^5 + 468466*x^4 - 389860*x^3 + 311630*x^2 - 163125*x + 96595, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 64 x^{18} - 291 x^{17} + 1069 x^{16} - 3248 x^{15} + 8514 x^{14} - 19384 x^{13} + 39326 x^{12} - 71194 x^{11} + 118256 x^{10} - 179424 x^{9} + 256621 x^{8} - 336594 x^{7} + 417836 x^{6} - 458653 x^{5} + 468466 x^{4} - 389860 x^{3} + 311630 x^{2} - 163125 x + 96595 \)

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:  \(3063157970528898000000000000000=2^{16}\cdot 3^{10}\cdot 5^{15}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.44$
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 Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{15} a^{12} - \frac{1}{15} a^{11} - \frac{2}{15} a^{10} - \frac{4}{15} a^{8} + \frac{1}{3} a^{7} - \frac{1}{5} a^{6} - \frac{2}{15} a^{5} - \frac{4}{15} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{15} a^{13} + \frac{2}{15} a^{11} - \frac{2}{15} a^{10} + \frac{2}{5} a^{9} + \frac{1}{15} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{5} - \frac{4}{15} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{165} a^{14} + \frac{4}{165} a^{13} + \frac{1}{55} a^{11} - \frac{6}{55} a^{10} - \frac{4}{33} a^{9} - \frac{12}{55} a^{8} - \frac{7}{15} a^{7} - \frac{4}{11} a^{6} - \frac{23}{55} a^{5} + \frac{67}{165} a^{4} + \frac{14}{33} a^{3} + \frac{1}{3} a^{2} - \frac{2}{11} a + \frac{14}{33}$, $\frac{1}{825} a^{15} - \frac{16}{825} a^{13} + \frac{14}{825} a^{12} + \frac{23}{275} a^{11} - \frac{1}{33} a^{10} + \frac{34}{75} a^{9} + \frac{23}{825} a^{8} + \frac{358}{825} a^{7} - \frac{9}{275} a^{6} - \frac{113}{275} a^{5} + \frac{11}{25} a^{4} - \frac{1}{165} a^{3} + \frac{38}{165} a^{2} - \frac{10}{33} a - \frac{23}{165}$, $\frac{1}{2475} a^{16} + \frac{1}{2475} a^{15} - \frac{2}{825} a^{14} + \frac{38}{2475} a^{13} - \frac{82}{2475} a^{12} - \frac{4}{275} a^{11} - \frac{326}{2475} a^{10} - \frac{353}{2475} a^{9} - \frac{16}{275} a^{8} + \frac{661}{2475} a^{7} - \frac{746}{2475} a^{6} + \frac{163}{825} a^{5} - \frac{118}{275} a^{4} + \frac{59}{165} a^{3} - \frac{59}{165} a^{2} - \frac{188}{495} a + \frac{7}{495}$, $\frac{1}{2475} a^{17} - \frac{1}{2475} a^{15} - \frac{1}{2475} a^{14} - \frac{2}{75} a^{13} - \frac{7}{495} a^{12} - \frac{1}{225} a^{11} + \frac{101}{825} a^{10} + \frac{1208}{2475} a^{9} + \frac{98}{225} a^{8} + \frac{247}{825} a^{7} + \frac{88}{225} a^{6} + \frac{23}{165} a^{5} - \frac{5}{33} a^{4} + \frac{52}{495} a^{2} + \frac{1}{3} a + \frac{10}{99}$, $\frac{1}{3769577447625} a^{18} - \frac{1}{418841938625} a^{17} - \frac{75945949}{3769577447625} a^{16} + \frac{55233436}{342688858875} a^{15} + \frac{620875972}{1256525815875} a^{14} - \frac{23670832556}{3769577447625} a^{13} - \frac{10796544911}{3769577447625} a^{12} + \frac{133381476688}{1256525815875} a^{11} + \frac{275571548969}{3769577447625} a^{10} + \frac{1856749862584}{3769577447625} a^{9} - \frac{34059778147}{1256525815875} a^{8} - \frac{1463169766621}{3769577447625} a^{7} + \frac{365654295653}{1256525815875} a^{6} + \frac{226882035748}{1256525815875} a^{5} - \frac{122395420607}{1256525815875} a^{4} - \frac{46085843272}{150783097905} a^{3} - \frac{12077032706}{83768387725} a^{2} - \frac{129970758331}{753915489525} a - \frac{71089919378}{251305163175}$, $\frac{1}{2642473790785125} a^{19} + \frac{31}{240224890071375} a^{18} + \frac{97958312912}{880824596928375} a^{17} + \frac{144609384286}{2642473790785125} a^{16} + \frac{1251716302711}{2642473790785125} a^{15} + \frac{927793698323}{880824596928375} a^{14} + \frac{32661061411634}{2642473790785125} a^{13} - \frac{21458910483031}{2642473790785125} a^{12} - \frac{52590179808407}{880824596928375} a^{11} - \frac{61223261857792}{880824596928375} a^{10} - \frac{858769791752146}{2642473790785125} a^{9} - \frac{373454458199327}{880824596928375} a^{8} + \frac{1066111733945729}{2642473790785125} a^{7} - \frac{63870893483394}{293608198976125} a^{6} + \frac{350985275023243}{880824596928375} a^{5} + \frac{10896287885363}{48044978014275} a^{4} + \frac{39360379792511}{528494758157025} a^{3} + \frac{48943540679743}{176164919385675} a^{2} - \frac{89286965776504}{528494758157025} a - \frac{6425944382581}{35232983877135}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( 7583540.84842 \) (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

\(\Q(\sqrt{5}) \), 4.0.136125.2, 5.1.2178000.1 x5, 10.2.23718420000000.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.2178000.1
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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$5.4.3.1$x^{4} - 5$$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}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$