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Properties

Label 20.16.4414566420...6624.1
Degree $20$
Signature $[16, 2]$
Discriminant $2^{10}\cdot 401^{11}$
Root discriminant $38.22$
Ramified primes $2, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T350

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 13, 13, -352, -335, -777, 674, 4737, 2368, -3568, -7832, 1454, 6529, -491, -2390, 103, 426, -8, -35, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 35*x^18 - 8*x^17 + 426*x^16 + 103*x^15 - 2390*x^14 - 491*x^13 + 6529*x^12 + 1454*x^11 - 7832*x^10 - 3568*x^9 + 2368*x^8 + 4737*x^7 + 674*x^6 - 777*x^5 - 335*x^4 - 352*x^3 + 13*x^2 + 13*x - 1)
 
gp: K = bnfinit(x^20 - 35*x^18 - 8*x^17 + 426*x^16 + 103*x^15 - 2390*x^14 - 491*x^13 + 6529*x^12 + 1454*x^11 - 7832*x^10 - 3568*x^9 + 2368*x^8 + 4737*x^7 + 674*x^6 - 777*x^5 - 335*x^4 - 352*x^3 + 13*x^2 + 13*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 35 x^{18} - 8 x^{17} + 426 x^{16} + 103 x^{15} - 2390 x^{14} - 491 x^{13} + 6529 x^{12} + 1454 x^{11} - 7832 x^{10} - 3568 x^{9} + 2368 x^{8} + 4737 x^{7} + 674 x^{6} - 777 x^{5} - 335 x^{4} - 352 x^{3} + 13 x^{2} + 13 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(44145664201696349530694455706624=2^{10}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.22$
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:  $2, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{13} a^{17} - \frac{4}{13} a^{16} - \frac{2}{13} a^{15} + \frac{5}{13} a^{14} + \frac{2}{13} a^{13} - \frac{5}{13} a^{12} + \frac{5}{13} a^{11} + \frac{5}{13} a^{10} + \frac{2}{13} a^{9} - \frac{2}{13} a^{8} - \frac{2}{13} a^{7} - \frac{3}{13} a^{6} + \frac{3}{13} a^{5} + \frac{4}{13} a^{4} - \frac{4}{13} a^{3} - \frac{6}{13} a^{2} + \frac{4}{13} a + \frac{5}{13}$, $\frac{1}{117} a^{18} - \frac{1}{39} a^{17} - \frac{2}{39} a^{16} - \frac{23}{117} a^{15} + \frac{11}{39} a^{14} - \frac{14}{39} a^{13} + \frac{4}{9} a^{12} - \frac{29}{117} a^{11} + \frac{11}{39} a^{10} - \frac{2}{9} a^{9} - \frac{56}{117} a^{8} - \frac{5}{117} a^{7} + \frac{1}{3} a^{6} + \frac{20}{117} a^{5} + \frac{2}{9} a^{4} - \frac{23}{117} a^{3} + \frac{11}{117} a^{2} - \frac{17}{117} a + \frac{5}{117}$, $\frac{1}{524318412167858465754831} a^{19} - \frac{618893892914054518504}{174772804055952821918277} a^{18} + \frac{1564654608987594295162}{174772804055952821918277} a^{17} - \frac{8236152193635664620722}{40332185551373728134987} a^{16} - \frac{45075213903530139052108}{174772804055952821918277} a^{15} + \frac{67012429934414167006867}{174772804055952821918277} a^{14} + \frac{210662234001718987420717}{524318412167858465754831} a^{13} + \frac{180241601742738251754919}{524318412167858465754831} a^{12} - \frac{61116941300085282619900}{174772804055952821918277} a^{11} + \frac{6795953523832208925593}{22796452702950368076297} a^{10} + \frac{161128813275498929919145}{524318412167858465754831} a^{9} + \frac{112322804402105432335921}{524318412167858465754831} a^{8} + \frac{41141439075859528501222}{174772804055952821918277} a^{7} + \frac{171646615017045451928288}{524318412167858465754831} a^{6} + \frac{213381209083328922265640}{524318412167858465754831} a^{5} - \frac{102581617512053927800823}{524318412167858465754831} a^{4} - \frac{218508063556894995746896}{524318412167858465754831} a^{3} + \frac{107374057731650691123670}{524318412167858465754831} a^{2} - \frac{16431574787924365493941}{524318412167858465754831} a - \frac{23981681827952474184883}{58257601351984273972759}$

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:  $17$
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:  \( 1174045114.07 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T350:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n350 are not computed
Character table for t20n350 is not computed

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.10.10.6$x^{10} - 5 x^{8} - 18 x^{6} - 46 x^{4} + 49 x^{2} - 13$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
401Data not computed