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Properties

Label 20.0.92561489592...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{15}\cdot 7^{10}$
Root discriminant $25.02$
Ramified primes $2, 5, 7$
Class number $4$ (GRH)
Class group $[2, 2]$ (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![25, 150, 1275, 850, -890, -2470, 6055, 3950, 1291, -7980, 5568, -1812, 1059, -1382, 1081, -494, 162, -54, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 21*x^18 - 54*x^17 + 162*x^16 - 494*x^15 + 1081*x^14 - 1382*x^13 + 1059*x^12 - 1812*x^11 + 5568*x^10 - 7980*x^9 + 1291*x^8 + 3950*x^7 + 6055*x^6 - 2470*x^5 - 890*x^4 + 850*x^3 + 1275*x^2 + 150*x + 25)
 
gp: K = bnfinit(x^20 - 6*x^19 + 21*x^18 - 54*x^17 + 162*x^16 - 494*x^15 + 1081*x^14 - 1382*x^13 + 1059*x^12 - 1812*x^11 + 5568*x^10 - 7980*x^9 + 1291*x^8 + 3950*x^7 + 6055*x^6 - 2470*x^5 - 890*x^4 + 850*x^3 + 1275*x^2 + 150*x + 25, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 21 x^{18} - 54 x^{17} + 162 x^{16} - 494 x^{15} + 1081 x^{14} - 1382 x^{13} + 1059 x^{12} - 1812 x^{11} + 5568 x^{10} - 7980 x^{9} + 1291 x^{8} + 3950 x^{7} + 6055 x^{6} - 2470 x^{5} - 890 x^{4} + 850 x^{3} + 1275 x^{2} + 150 x + 25 \)

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:  \(9256148959232000000000000000=2^{30}\cdot 5^{15}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.02$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{20} a^{12} - \frac{1}{10} a^{11} + \frac{1}{10} a^{10} - \frac{1}{5} a^{8} - \frac{1}{10} a^{6} - \frac{3}{10} a^{5} + \frac{3}{10} a^{4} + \frac{1}{4}$, $\frac{1}{20} a^{13} - \frac{1}{10} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{10} a^{8} - \frac{1}{10} a^{7} - \frac{3}{10} a^{5} - \frac{2}{5} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{20} a^{14} + \frac{1}{10} a^{9} - \frac{2}{5} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{40} a^{15} - \frac{1}{40} a^{12} + \frac{1}{20} a^{11} - \frac{1}{4} a^{9} - \frac{3}{20} a^{8} + \frac{1}{20} a^{6} - \frac{1}{20} a^{5} + \frac{1}{10} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{120} a^{16} - \frac{1}{120} a^{13} - \frac{1}{60} a^{12} + \frac{7}{30} a^{11} + \frac{1}{60} a^{10} + \frac{7}{60} a^{9} - \frac{1}{5} a^{8} + \frac{1}{60} a^{7} + \frac{1}{20} a^{6} + \frac{1}{15} a^{5} - \frac{19}{120} a^{4} + \frac{1}{12} a^{3} + \frac{1}{6} a^{2} - \frac{5}{24} a + \frac{1}{6}$, $\frac{1}{16680} a^{17} - \frac{13}{3336} a^{16} + \frac{1}{139} a^{15} + \frac{7}{3336} a^{14} - \frac{7}{5560} a^{13} + \frac{23}{4170} a^{12} - \frac{613}{2780} a^{11} + \frac{919}{4170} a^{10} - \frac{1613}{8340} a^{9} + \frac{1129}{8340} a^{8} - \frac{499}{4170} a^{7} + \frac{191}{1668} a^{6} + \frac{5881}{16680} a^{5} - \frac{1969}{5560} a^{4} + \frac{177}{556} a^{3} - \frac{181}{3336} a^{2} - \frac{1003}{3336} a + \frac{317}{1668}$, $\frac{1}{2129035200} a^{18} - \frac{95}{7096784} a^{17} - \frac{1427}{177419600} a^{16} - \frac{5631929}{1064517600} a^{15} - \frac{19671283}{1064517600} a^{14} - \frac{259709}{53225880} a^{13} + \frac{40560791}{2129035200} a^{12} + \frac{33506833}{266129400} a^{11} + \frac{42610329}{177419600} a^{10} - \frac{8292541}{532258800} a^{9} - \frac{33296983}{177419600} a^{8} - \frac{848611}{133064700} a^{7} - \frac{8019673}{85161408} a^{6} - \frac{8416493}{21290352} a^{5} - \frac{6406061}{26612940} a^{4} + \frac{58266167}{212903520} a^{3} - \frac{17753891}{42580704} a^{2} + \frac{204229}{1774196} a - \frac{9150197}{28387136}$, $\frac{1}{32357110740356342400} a^{19} + \frac{278321153}{32357110740356342400} a^{18} - \frac{3863830895897}{168526618439355950} a^{17} + \frac{6192550900934657}{3235711074035634240} a^{16} + \frac{3637371957235913}{808927768508908560} a^{15} + \frac{1966599323584027}{5392851790059390400} a^{14} - \frac{125745641455211969}{32357110740356342400} a^{13} - \frac{87268714993285951}{10785703580118780800} a^{12} - \frac{223861650188034511}{1617855537017817120} a^{11} - \frac{29553912467296433}{161785553701781712} a^{10} + \frac{729116592865087559}{4044638842544542800} a^{9} - \frac{438307710881105941}{8089277685089085600} a^{8} - \frac{1728011920000988731}{10785703580118780800} a^{7} - \frac{70016308170559829}{6471422148071268480} a^{6} - \frac{634686174066119107}{1617855537017817120} a^{5} + \frac{524241032979983829}{1078570358011878080} a^{4} + \frac{51141786247664429}{134821294751484760} a^{3} + \frac{254825111246154485}{647142214807126848} a^{2} - \frac{194911892439509445}{431428143204751232} a - \frac{85350657992609233}{431428143204751232}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 905204.455446 \) (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

\(\Q(\sqrt{5}) \), 4.0.392000.2, 5.1.392000.1 x5, 10.2.768320000000.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.392000.1
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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.4$x^{4} - 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$7$7.4.2.2$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}$
7.4.2.2$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}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$