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Properties

Label 20.20.2911964953...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{16}\cdot 5^{10}\cdot 13^{4}\cdot 29^{7}\cdot 31^{4}$
Root discriminant $42.00$
Ramified primes $2, 5, 13, 29, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T658

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 17, -173, -465, 3797, 5754, -23634, -30246, 45218, 45384, -39010, -27300, 17285, 8051, -4161, -1245, 548, 97, -37, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 37*x^18 + 97*x^17 + 548*x^16 - 1245*x^15 - 4161*x^14 + 8051*x^13 + 17285*x^12 - 27300*x^11 - 39010*x^10 + 45384*x^9 + 45218*x^8 - 30246*x^7 - 23634*x^6 + 5754*x^5 + 3797*x^4 - 465*x^3 - 173*x^2 + 17*x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 - 37*x^18 + 97*x^17 + 548*x^16 - 1245*x^15 - 4161*x^14 + 8051*x^13 + 17285*x^12 - 27300*x^11 - 39010*x^10 + 45384*x^9 + 45218*x^8 - 30246*x^7 - 23634*x^6 + 5754*x^5 + 3797*x^4 - 465*x^3 - 173*x^2 + 17*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 37 x^{18} + 97 x^{17} + 548 x^{16} - 1245 x^{15} - 4161 x^{14} + 8051 x^{13} + 17285 x^{12} - 27300 x^{11} - 39010 x^{10} + 45384 x^{9} + 45218 x^{8} - 30246 x^{7} - 23634 x^{6} + 5754 x^{5} + 3797 x^{4} - 465 x^{3} - 173 x^{2} + 17 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(291196495384907705490560000000000=2^{16}\cdot 5^{10}\cdot 13^{4}\cdot 29^{7}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.00$
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:  $2, 5, 13, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{29} a^{17} + \frac{12}{29} a^{16} + \frac{10}{29} a^{15} - \frac{6}{29} a^{14} - \frac{10}{29} a^{13} - \frac{14}{29} a^{12} + \frac{8}{29} a^{11} - \frac{4}{29} a^{10} - \frac{2}{29} a^{9} + \frac{12}{29} a^{8} - \frac{1}{29} a^{7} - \frac{6}{29} a^{6} + \frac{13}{29} a^{5} - \frac{1}{29} a^{4} + \frac{1}{29} a^{3} - \frac{13}{29} a^{2} + \frac{9}{29} a - \frac{13}{29}$, $\frac{1}{377} a^{18} + \frac{6}{377} a^{17} - \frac{7}{29} a^{16} + \frac{21}{377} a^{15} + \frac{113}{377} a^{14} + \frac{8}{29} a^{13} - \frac{24}{377} a^{12} + \frac{35}{377} a^{11} - \frac{36}{377} a^{10} + \frac{82}{377} a^{9} + \frac{14}{377} a^{8} - \frac{2}{13} a^{7} - \frac{67}{377} a^{6} + \frac{37}{377} a^{5} + \frac{94}{377} a^{4} - \frac{48}{377} a^{3} + \frac{4}{13} a^{2} - \frac{9}{377} a + \frac{49}{377}$, $\frac{1}{39740477752978215301550033} a^{19} - \frac{2693044003309954025503}{39740477752978215301550033} a^{18} + \frac{453812255432504441879873}{39740477752978215301550033} a^{17} + \frac{16740474688213101284392123}{39740477752978215301550033} a^{16} + \frac{16613560917066096463228475}{39740477752978215301550033} a^{15} - \frac{10974690084065987838374278}{39740477752978215301550033} a^{14} - \frac{5022993629730612840443891}{39740477752978215301550033} a^{13} + \frac{709027556865088092562230}{3056959827152170407811541} a^{12} - \frac{65801505894631979819183}{39740477752978215301550033} a^{11} - \frac{7594528536581485021506818}{39740477752978215301550033} a^{10} + \frac{17214467773296478672960904}{39740477752978215301550033} a^{9} + \frac{19533819404132683031366617}{39740477752978215301550033} a^{8} + \frac{16062221866228495030771723}{39740477752978215301550033} a^{7} + \frac{4077983805371834350024897}{39740477752978215301550033} a^{6} + \frac{7677467369918974897884055}{39740477752978215301550033} a^{5} - \frac{12308827120671921540067695}{39740477752978215301550033} a^{4} - \frac{3237134193817245232556987}{39740477752978215301550033} a^{3} - \frac{4822235566323246805110310}{39740477752978215301550033} a^{2} + \frac{15826117911245728396071403}{39740477752978215301550033} a + \frac{2456764729326857260272345}{39740477752978215301550033}$

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:  $19$
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:  \( 7000577124.85 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T658:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 76 conjugacy class representatives for t20n658 are not computed
Character table for t20n658 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 10.10.109268775200000.1

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 24 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 $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.6.5.2$x^{6} + 58$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
29.10.0.1$x^{10} + x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.6.4.3$x^{6} + 713 x^{3} + 138384$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$