Properties

Label 20.0.25000000000...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{22}$
Root discriminant $11.75$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $C_{10}\times D_5$ (as 20T24)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 7*x^18 - 6*x^17 - 2*x^16 + 6*x^15 + 14*x^14 - 54*x^13 + 64*x^12 + 24*x^11 - 190*x^10 + 290*x^9 - 169*x^8 - 154*x^7 + 464*x^6 - 554*x^5 + 417*x^4 - 208*x^3 + 66*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 7*x^18 - 6*x^17 - 2*x^16 + 6*x^15 + 14*x^14 - 54*x^13 + 64*x^12 + 24*x^11 - 190*x^10 + 290*x^9 - 169*x^8 - 154*x^7 + 464*x^6 - 554*x^5 + 417*x^4 - 208*x^3 + 66*x^2 - 12*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -12, 66, -208, 417, -554, 464, -154, -169, 290, -190, 24, 64, -54, 14, 6, -2, -6, 7, -4, 1]);
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 7 x^{18} - 6 x^{17} - 2 x^{16} + 6 x^{15} + 14 x^{14} - 54 x^{13} + 64 x^{12} + 24 x^{11} - 190 x^{10} + 290 x^{9} - 169 x^{8} - 154 x^{7} + 464 x^{6} - 554 x^{5} + 417 x^{4} - 208 x^{3} + 66 x^{2} - 12 x + 1 \)

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(2500000000000000000000=2^{20}\cdot 5^{22}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $11.75$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 5$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $10$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} - \frac{1}{7} a^{15} - \frac{2}{7} a^{14} - \frac{3}{7} a^{13} + \frac{2}{7} a^{11} - \frac{2}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{3} - \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{17} - \frac{3}{7} a^{15} + \frac{2}{7} a^{14} - \frac{3}{7} a^{13} + \frac{2}{7} a^{12} + \frac{2}{7} a^{11} - \frac{2}{7} a^{10} - \frac{1}{7} a^{9} + \frac{2}{7} a^{8} - \frac{2}{7} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} + \frac{3}{7} a^{4} - \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{18} - \frac{1}{7} a^{15} - \frac{2}{7} a^{14} + \frac{2}{7} a^{12} - \frac{3}{7} a^{11} - \frac{1}{7} a^{10} + \frac{3}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{6} - \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{2}{7} a^{3} + \frac{1}{7} a^{2} - \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{22230187} a^{19} + \frac{316742}{22230187} a^{18} - \frac{1248133}{22230187} a^{17} + \frac{72912}{3175741} a^{16} - \frac{5356625}{22230187} a^{15} - \frac{2276879}{22230187} a^{14} + \frac{369525}{3175741} a^{13} + \frac{9538906}{22230187} a^{12} + \frac{10332540}{22230187} a^{11} + \frac{1228161}{3175741} a^{10} - \frac{222459}{22230187} a^{9} - \frac{6008298}{22230187} a^{8} + \frac{9247629}{22230187} a^{7} - \frac{1567752}{22230187} a^{6} - \frac{4433063}{22230187} a^{5} + \frac{8436821}{22230187} a^{4} + \frac{9943944}{22230187} a^{3} + \frac{1386475}{22230187} a^{2} - \frac{109510}{22230187} a - \frac{201610}{3175741}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{57119}{9107} a^{19} + \frac{26866}{1301} a^{18} - \frac{267410}{9107} a^{17} + \frac{156983}{9107} a^{16} + \frac{218307}{9107} a^{15} - \frac{181352}{9107} a^{14} - \frac{926865}{9107} a^{13} + \frac{2419212}{9107} a^{12} - \frac{1951750}{9107} a^{11} - \frac{2701613}{9107} a^{10} + \frac{1266459}{1301} a^{9} - \frac{10294686}{9107} a^{8} + \frac{2557279}{9107} a^{7} + \frac{10292841}{9107} a^{6} - \frac{19014662}{9107} a^{5} + \frac{18350921}{9107} a^{4} - \frac{11338829}{9107} a^{3} + \frac{4435672}{9107} a^{2} - \frac{1011806}{9107} a + \frac{107027}{9107} \) (order $4$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 397.737246993 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$C_{10}\times D_5$ (as 20T24):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 100
The 40 conjugacy class representatives for $C_{10}\times D_5$
Character table for $C_{10}\times D_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), \(\Q(i, \sqrt{5})\), 10.0.400000000.1

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

Sibling fields

Degree 20 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$5$5.10.17.5$x^{10} - 5 x^{8} + 55$$10$$1$$17$$C_{10}$$[2]_{2}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$