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Properties

Label 20.10.1215000000...0000.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{30}\cdot 3^{5}\cdot 5^{31}$
Root discriminant $45.11$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times D_5\wr C_2$ (as 20T92)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, 60, -565, 3470, -13315, 31812, -45840, 32500, 7320, -37080, 31089, -8710, -2795, 3380, -2250, 1348, -500, 60, 25, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 25*x^18 + 60*x^17 - 500*x^16 + 1348*x^15 - 2250*x^14 + 3380*x^13 - 2795*x^12 - 8710*x^11 + 31089*x^10 - 37080*x^9 + 7320*x^8 + 32500*x^7 - 45840*x^6 + 31812*x^5 - 13315*x^4 + 3470*x^3 - 565*x^2 + 60*x - 4)
 
gp: K = bnfinit(x^20 - 10*x^19 + 25*x^18 + 60*x^17 - 500*x^16 + 1348*x^15 - 2250*x^14 + 3380*x^13 - 2795*x^12 - 8710*x^11 + 31089*x^10 - 37080*x^9 + 7320*x^8 + 32500*x^7 - 45840*x^6 + 31812*x^5 - 13315*x^4 + 3470*x^3 - 565*x^2 + 60*x - 4, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 25 x^{18} + 60 x^{17} - 500 x^{16} + 1348 x^{15} - 2250 x^{14} + 3380 x^{13} - 2795 x^{12} - 8710 x^{11} + 31089 x^{10} - 37080 x^{9} + 7320 x^{8} + 32500 x^{7} - 45840 x^{6} + 31812 x^{5} - 13315 x^{4} + 3470 x^{3} - 565 x^{2} + 60 x - 4 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1215000000000000000000000000000000=-\,2^{30}\cdot 3^{5}\cdot 5^{31}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{3}{8} a^{5} - \frac{3}{8} a^{4} + \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{2208} a^{16} - \frac{1}{276} a^{15} - \frac{37}{1104} a^{14} + \frac{53}{1104} a^{13} + \frac{229}{2208} a^{12} + \frac{49}{552} a^{11} + \frac{15}{736} a^{10} - \frac{33}{368} a^{9} - \frac{467}{2208} a^{8} - \frac{17}{184} a^{7} - \frac{383}{2208} a^{6} - \frac{175}{368} a^{5} + \frac{27}{184} a^{4} - \frac{55}{138} a^{3} + \frac{141}{736} a^{2} - \frac{67}{552} a - \frac{1}{552}$, $\frac{1}{2208} a^{17} - \frac{1}{16} a^{15} + \frac{11}{368} a^{14} - \frac{9}{736} a^{13} - \frac{15}{184} a^{12} - \frac{43}{2208} a^{11} + \frac{27}{368} a^{10} - \frac{395}{2208} a^{9} + \frac{119}{552} a^{8} - \frac{359}{2208} a^{7} + \frac{151}{1104} a^{6} - \frac{75}{184} a^{5} + \frac{19}{69} a^{4} - \frac{545}{2208} a^{3} - \frac{187}{552} a^{2} + \frac{51}{184} a + \frac{67}{138}$, $\frac{1}{98551872} a^{18} - \frac{1}{10950208} a^{17} - \frac{3733}{98551872} a^{16} + \frac{7517}{24637968} a^{15} - \frac{3650479}{98551872} a^{14} + \frac{388481}{98551872} a^{13} - \frac{122509}{1145952} a^{12} + \frac{10259201}{98551872} a^{11} + \frac{1811881}{24637968} a^{10} + \frac{15003257}{98551872} a^{9} + \frac{2130445}{16425312} a^{8} - \frac{9550711}{98551872} a^{7} + \frac{14248115}{98551872} a^{6} + \frac{2430205}{49275936} a^{5} + \frac{47130307}{98551872} a^{4} + \frac{13123559}{32850624} a^{3} - \frac{11094383}{32850624} a^{2} + \frac{531877}{12318984} a - \frac{4943909}{24637968}$, $\frac{1}{98551872} a^{19} - \frac{1907}{49275936} a^{17} - \frac{3529}{98551872} a^{16} - \frac{3379867}{98551872} a^{15} + \frac{2245561}{49275936} a^{14} + \frac{5279539}{98551872} a^{13} - \frac{10648861}{98551872} a^{12} - \frac{11290523}{98551872} a^{11} + \frac{6317069}{98551872} a^{10} - \frac{4111603}{32850624} a^{9} - \frac{411547}{2291904} a^{8} - \frac{2528341}{24637968} a^{7} - \frac{14734363}{98551872} a^{6} - \frac{19996859}{98551872} a^{5} - \frac{14155}{47748} a^{4} - \frac{1508233}{4106328} a^{3} - \frac{36594661}{98551872} a^{2} - \frac{334309}{1071216} a - \frac{837581}{2737552}$

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:  $14$
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:  \( 15994113224.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_5\wr C_2$ (as 20T92):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $C_2\times D_5\wr C_2$
Character table for $C_2\times D_5\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{10}) \), 4.2.24000.2, 10.6.9000000000000000.2

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.10.0.1$x^{10} - x^{3} - x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
5Data not computed