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Properties

Label 20.4.14862445601...5568.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{10}\cdot 3^{10}\cdot 7^{3}\cdot 19^{10}\cdot 43^{8}$
Root discriminant $64.36$
Ramified primes $2, 3, 7, 19, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T525

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![343, 0, 15827, 0, 132013, 0, 236114, 0, 19869, 0, -41203, 0, -11818, 0, -329, 0, 233, 0, 29, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 29*x^18 + 233*x^16 - 329*x^14 - 11818*x^12 - 41203*x^10 + 19869*x^8 + 236114*x^6 + 132013*x^4 + 15827*x^2 + 343)
 
gp: K = bnfinit(x^20 + 29*x^18 + 233*x^16 - 329*x^14 - 11818*x^12 - 41203*x^10 + 19869*x^8 + 236114*x^6 + 132013*x^4 + 15827*x^2 + 343, 1)
 

Normalized defining polynomial

\( x^{20} + 29 x^{18} + 233 x^{16} - 329 x^{14} - 11818 x^{12} - 41203 x^{10} + 19869 x^{8} + 236114 x^{6} + 132013 x^{4} + 15827 x^{2} + 343 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1486244560113347729221539626477165568=2^{10}\cdot 3^{10}\cdot 7^{3}\cdot 19^{10}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.36$
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:  $2, 3, 7, 19, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{35} a^{14} + \frac{1}{7} a^{12} - \frac{6}{35} a^{10} - \frac{3}{35} a^{8} - \frac{1}{5} a^{6} - \frac{3}{7} a^{4} + \frac{6}{35} a^{2} - \frac{2}{5}$, $\frac{1}{70} a^{15} - \frac{1}{70} a^{14} + \frac{1}{14} a^{13} + \frac{3}{7} a^{12} + \frac{29}{70} a^{11} - \frac{29}{70} a^{10} - \frac{3}{70} a^{9} + \frac{3}{70} a^{8} - \frac{1}{10} a^{7} - \frac{2}{5} a^{6} + \frac{2}{7} a^{5} - \frac{2}{7} a^{4} - \frac{29}{70} a^{3} - \frac{3}{35} a^{2} + \frac{3}{10} a - \frac{3}{10}$, $\frac{1}{3010} a^{16} + \frac{8}{1505} a^{14} - \frac{1}{2} a^{13} - \frac{133}{430} a^{12} + \frac{88}{1505} a^{10} + \frac{45}{301} a^{8} - \frac{1}{2} a^{7} - \frac{606}{1505} a^{6} + \frac{681}{3010} a^{4} - \frac{1}{2} a^{3} - \frac{11}{70} a^{2} - \frac{137}{430}$, $\frac{1}{3010} a^{17} + \frac{8}{1505} a^{15} - \frac{1}{70} a^{14} - \frac{133}{430} a^{13} + \frac{3}{7} a^{12} + \frac{88}{1505} a^{11} + \frac{3}{35} a^{10} + \frac{45}{301} a^{9} + \frac{3}{70} a^{8} - \frac{606}{1505} a^{7} - \frac{2}{5} a^{6} + \frac{681}{3010} a^{5} + \frac{3}{14} a^{4} - \frac{11}{70} a^{3} - \frac{3}{35} a^{2} - \frac{137}{430} a + \frac{1}{5}$, $\frac{1}{20216880866090090} a^{18} - \frac{1535483632903}{10108440433045045} a^{16} + \frac{2301804453453}{235080010070815} a^{14} - \frac{1}{2} a^{13} - \frac{135755499628698}{1444062919006435} a^{12} - \frac{1}{2} a^{11} - \frac{770274391567091}{1555144682006930} a^{10} - \frac{1486311770585001}{4043376173218018} a^{8} - \frac{1}{2} a^{7} - \frac{2953305470433567}{20216880866090090} a^{6} - \frac{1}{2} a^{5} + \frac{665961869124515}{4043376173218018} a^{4} - \frac{1}{2} a^{3} - \frac{1175471710806781}{2888125838012870} a^{2} - \frac{1}{2} a + \frac{139933587156721}{412589405430410}$, $\frac{1}{20216880866090090} a^{19} - \frac{1535483632903}{10108440433045045} a^{17} - \frac{2112962809403}{470160020141630} a^{15} - \frac{477805701972601}{2888125838012870} a^{13} - \frac{1}{2} a^{12} + \frac{70298032518484}{777572341003465} a^{11} - \frac{1}{2} a^{10} - \frac{3282560550760572}{10108440433045045} a^{9} - \frac{465808691912279}{10108440433045045} a^{7} - \frac{1}{2} a^{6} - \frac{489288466080633}{4043376173218018} a^{5} - \frac{1}{2} a^{4} + \frac{10518782470704}{1444062919006435} a^{3} - \frac{1}{2} a^{2} + \frac{8078382763799}{206294702715205} a - \frac{1}{2}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 21073818520.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T525:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 152 conjugacy class representatives for t20n525 are not computed
Character table for t20n525 is not computed

Intermediate fields

\(\Q(\sqrt{57}) \), 5.5.667489.1, 10.10.2057065406163657.1

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 ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ R $20$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ $20$ R $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$2.10.10.14$x^{10} + 5 x^{8} - 50 x^{6} - 58 x^{4} + 49 x^{2} + 21$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$