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Properties

Label 20.0.37791360000...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 3^{10}\cdot 5^{22}$
Root discriminant $26.85$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![361, 0, 3350, 0, 30425, 0, -36160, 0, 17350, 0, -5772, 0, 1750, 0, -400, 0, 65, 0, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^18 + 65*x^16 - 400*x^14 + 1750*x^12 - 5772*x^10 + 17350*x^8 - 36160*x^6 + 30425*x^4 + 3350*x^2 + 361)
 
gp: K = bnfinit(x^20 - 10*x^18 + 65*x^16 - 400*x^14 + 1750*x^12 - 5772*x^10 + 17350*x^8 - 36160*x^6 + 30425*x^4 + 3350*x^2 + 361, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{18} + 65 x^{16} - 400 x^{14} + 1750 x^{12} - 5772 x^{10} + 17350 x^{8} - 36160 x^{6} + 30425 x^{4} + 3350 x^{2} + 361 \)

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:  \(37791360000000000000000000000=2^{28}\cdot 3^{10}\cdot 5^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.85$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{10} - \frac{1}{6} a^{6} - \frac{1}{4} a^{4} - \frac{1}{12} a^{2} - \frac{1}{6}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{12} + \frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{8} a^{5} + \frac{3}{8} a^{4} - \frac{5}{12} a^{3} + \frac{5}{12} a^{2} - \frac{5}{24} a - \frac{7}{24}$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{12} - \frac{1}{8} a^{10} - \frac{5}{24} a^{8} + \frac{1}{8} a^{6} - \frac{7}{24} a^{4} + \frac{7}{24} a^{2} - \frac{1}{8}$, $\frac{1}{24} a^{15} - \frac{1}{24} a^{12} - \frac{1}{24} a^{11} - \frac{1}{12} a^{10} - \frac{1}{12} a^{9} - \frac{1}{8} a^{8} - \frac{5}{24} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{3}{8} a^{4} + \frac{3}{8} a^{3} + \frac{5}{12} a^{2} + \frac{5}{12} a - \frac{1}{24}$, $\frac{1}{72} a^{16} - \frac{1}{72} a^{14} + \frac{1}{72} a^{12} + \frac{1}{24} a^{10} + \frac{1}{24} a^{8} - \frac{1}{24} a^{6} - \frac{17}{72} a^{4} - \frac{7}{72} a^{2} - \frac{5}{18}$, $\frac{1}{144} a^{17} - \frac{1}{144} a^{16} + \frac{1}{72} a^{15} - \frac{1}{72} a^{14} - \frac{1}{72} a^{13} - \frac{1}{36} a^{12} - \frac{1}{24} a^{11} - \frac{1}{24} a^{10} - \frac{1}{12} a^{9} - \frac{1}{24} a^{8} - \frac{5}{24} a^{7} + \frac{1}{24} a^{6} - \frac{1}{72} a^{5} - \frac{1}{9} a^{4} + \frac{25}{72} a^{3} + \frac{5}{72} a^{2} - \frac{65}{144} a - \frac{13}{144}$, $\frac{1}{44338212144} a^{18} - \frac{103286291}{44338212144} a^{16} - \frac{59520871}{11084553036} a^{14} + \frac{39674291}{2771138259} a^{12} - \frac{39870397}{2463234008} a^{10} + \frac{100988213}{7389702024} a^{8} - \frac{2057274053}{11084553036} a^{6} - \frac{648930823}{5542276518} a^{4} + \frac{19865967209}{44338212144} a^{2} - \frac{8744816611}{44338212144}$, $\frac{1}{842426030736} a^{19} + \frac{166740748}{52651626921} a^{17} - \frac{1}{144} a^{16} + \frac{2652096517}{421213015368} a^{15} - \frac{1}{72} a^{14} - \frac{7996020449}{421213015368} a^{13} - \frac{1}{36} a^{12} + \frac{836769091}{11700361538} a^{11} - \frac{1}{24} a^{10} - \frac{3593862799}{140404338456} a^{9} + \frac{5}{24} a^{8} - \frac{1343409847}{421213015368} a^{7} + \frac{1}{24} a^{6} - \frac{66331903249}{421213015368} a^{5} + \frac{7}{18} a^{4} - \frac{96521839669}{842426030736} a^{3} + \frac{5}{72} a^{2} + \frac{210390806767}{421213015368} a + \frac{23}{144}$

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:  \( \frac{5025}{48549218} a^{19} - \frac{227325}{97098436} a^{17} + \frac{382316}{24274609} a^{15} - \frac{4775355}{48549218} a^{13} + \frac{51157935}{97098436} a^{11} - \frac{175527875}{97098436} a^{9} + \frac{134281725}{24274609} a^{7} - \frac{707859771}{48549218} a^{5} + \frac{1536315655}{97098436} a^{3} + \frac{85556970}{24274609} a \) (order $4$)
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:  \( 5376688.032388041 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{15}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{15})\), 5.1.50000.1, 10.2.194400000000000.1, 10.0.3037500000000.2, 10.0.160000000000.1

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

Sibling fields

Degree 20 siblings: 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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{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$$[\ ]_{2}^{4}$
$5$5.10.11.7$x^{10} + 5 x^{2} + 10$$10$$1$$11$$F_{5}\times C_2$$[5/4]_{4}^{2}$
5.10.11.7$x^{10} + 5 x^{2} + 10$$10$$1$$11$$F_{5}\times C_2$$[5/4]_{4}^{2}$