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Properties

Label 20.6.83510262074...0187.2
Degree $20$
Signature $[6, 7]$
Discriminant $-\,13^{10}\cdot 347^{7}$
Root discriminant $27.93$
Ramified primes $13, 347$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T375

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![53, 519, -1298, -1169, 5180, -5117, 1909, 1444, -2876, 2610, -1643, 869, -311, -28, 127, -153, 89, -46, 17, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 17*x^18 - 46*x^17 + 89*x^16 - 153*x^15 + 127*x^14 - 28*x^13 - 311*x^12 + 869*x^11 - 1643*x^10 + 2610*x^9 - 2876*x^8 + 1444*x^7 + 1909*x^6 - 5117*x^5 + 5180*x^4 - 1169*x^3 - 1298*x^2 + 519*x + 53)
 
gp: K = bnfinit(x^20 - 4*x^19 + 17*x^18 - 46*x^17 + 89*x^16 - 153*x^15 + 127*x^14 - 28*x^13 - 311*x^12 + 869*x^11 - 1643*x^10 + 2610*x^9 - 2876*x^8 + 1444*x^7 + 1909*x^6 - 5117*x^5 + 5180*x^4 - 1169*x^3 - 1298*x^2 + 519*x + 53, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 17 x^{18} - 46 x^{17} + 89 x^{16} - 153 x^{15} + 127 x^{14} - 28 x^{13} - 311 x^{12} + 869 x^{11} - 1643 x^{10} + 2610 x^{9} - 2876 x^{8} + 1444 x^{7} + 1909 x^{6} - 5117 x^{5} + 5180 x^{4} - 1169 x^{3} - 1298 x^{2} + 519 x + 53 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-83510262074750967216189850187=-\,13^{10}\cdot 347^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.93$
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:  $13, 347$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} - \frac{1}{13} a^{12} + \frac{1}{13} a^{11} - \frac{6}{13} a^{10} + \frac{3}{13} a^{9} + \frac{2}{13} a^{8} - \frac{5}{13} a^{7} + \frac{6}{13} a^{6} - \frac{3}{13} a^{5} - \frac{2}{13} a^{4} - \frac{2}{13} a^{3} - \frac{2}{13} a^{2} + \frac{2}{13} a + \frac{4}{13}$, $\frac{1}{13} a^{15} - \frac{1}{13} a^{13} + \frac{1}{13} a^{12} - \frac{6}{13} a^{11} + \frac{3}{13} a^{10} + \frac{2}{13} a^{9} - \frac{5}{13} a^{8} + \frac{6}{13} a^{7} - \frac{3}{13} a^{6} - \frac{2}{13} a^{5} - \frac{2}{13} a^{4} - \frac{2}{13} a^{3} + \frac{2}{13} a^{2} + \frac{4}{13} a$, $\frac{1}{13} a^{16} + \frac{1}{13} a^{13} + \frac{6}{13} a^{12} + \frac{4}{13} a^{11} - \frac{4}{13} a^{10} - \frac{2}{13} a^{9} - \frac{5}{13} a^{8} + \frac{5}{13} a^{7} + \frac{4}{13} a^{6} - \frac{5}{13} a^{5} - \frac{4}{13} a^{4} + \frac{2}{13} a^{2} + \frac{2}{13} a + \frac{4}{13}$, $\frac{1}{13} a^{17} + \frac{6}{13} a^{13} + \frac{5}{13} a^{12} - \frac{5}{13} a^{11} + \frac{4}{13} a^{10} + \frac{5}{13} a^{9} + \frac{3}{13} a^{8} - \frac{4}{13} a^{7} + \frac{2}{13} a^{6} - \frac{1}{13} a^{5} + \frac{2}{13} a^{4} + \frac{4}{13} a^{3} + \frac{4}{13} a^{2} + \frac{2}{13} a - \frac{4}{13}$, $\frac{1}{13} a^{18} + \frac{5}{13} a^{13} + \frac{1}{13} a^{12} - \frac{2}{13} a^{11} + \frac{2}{13} a^{10} - \frac{2}{13} a^{9} - \frac{3}{13} a^{8} + \frac{6}{13} a^{7} + \frac{2}{13} a^{6} - \frac{6}{13} a^{5} + \frac{3}{13} a^{4} + \frac{3}{13} a^{3} + \frac{1}{13} a^{2} - \frac{3}{13} a + \frac{2}{13}$, $\frac{1}{22200418964458914435864820381} a^{19} + \frac{803180499557895544377431640}{22200418964458914435864820381} a^{18} + \frac{127034707404942755688836083}{22200418964458914435864820381} a^{17} + \frac{609463645215217328038828758}{22200418964458914435864820381} a^{16} - \frac{851475932502461700383785352}{22200418964458914435864820381} a^{15} + \frac{688360439416926537159418400}{22200418964458914435864820381} a^{14} + \frac{3821872196160266443140214320}{22200418964458914435864820381} a^{13} + \frac{2449001966775344884589178614}{22200418964458914435864820381} a^{12} - \frac{8898671631625295845228576009}{22200418964458914435864820381} a^{11} - \frac{10369901269019226395667097587}{22200418964458914435864820381} a^{10} + \frac{1499220901582410860849668542}{22200418964458914435864820381} a^{9} - \frac{8744290590008291994634547180}{22200418964458914435864820381} a^{8} - \frac{25428416880916663249872968}{1707724535727608802758832337} a^{7} + \frac{5042559782090395125059973359}{22200418964458914435864820381} a^{6} + \frac{723342680681785821376635578}{1707724535727608802758832337} a^{5} + \frac{2689636511657074083932614398}{22200418964458914435864820381} a^{4} - \frac{4173210163366541400149874193}{22200418964458914435864820381} a^{3} + \frac{757769226088635974744656419}{1707724535727608802758832337} a^{2} + \frac{7689707542736277498353498436}{22200418964458914435864820381} a + \frac{8030690296571916444083580701}{22200418964458914435864820381}$

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:  $12$
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:  \( 4641956.03417 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T375:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 48 conjugacy class representatives for t20n375
Character table for t20n375 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.3.4511.1, 10.6.44707018837.1

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
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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$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}$
13.4.2.1$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}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
347Data not computed