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Properties

Label 20.6.78574805586...9483.3
Degree $20$
Signature $[6, 7]$
Discriminant $-\,13^{10}\cdot 97^{2}\cdot 347^{7}$
Root discriminant $44.13$
Ramified primes $13, 97, 347$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T796

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-564811, -1209234, -1818712, -2059358, -353620, 777200, 80199, -171479, 14292, 8727, 13020, 6806, -7101, -490, 1415, -98, -214, 40, 15, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 15*x^18 + 40*x^17 - 214*x^16 - 98*x^15 + 1415*x^14 - 490*x^13 - 7101*x^12 + 6806*x^11 + 13020*x^10 + 8727*x^9 + 14292*x^8 - 171479*x^7 + 80199*x^6 + 777200*x^5 - 353620*x^4 - 2059358*x^3 - 1818712*x^2 - 1209234*x - 564811)
 
gp: K = bnfinit(x^20 - 7*x^19 + 15*x^18 + 40*x^17 - 214*x^16 - 98*x^15 + 1415*x^14 - 490*x^13 - 7101*x^12 + 6806*x^11 + 13020*x^10 + 8727*x^9 + 14292*x^8 - 171479*x^7 + 80199*x^6 + 777200*x^5 - 353620*x^4 - 2059358*x^3 - 1818712*x^2 - 1209234*x - 564811, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 15 x^{18} + 40 x^{17} - 214 x^{16} - 98 x^{15} + 1415 x^{14} - 490 x^{13} - 7101 x^{12} + 6806 x^{11} + 13020 x^{10} + 8727 x^{9} + 14292 x^{8} - 171479 x^{7} + 80199 x^{6} + 777200 x^{5} - 353620 x^{4} - 2059358 x^{3} - 1818712 x^{2} - 1209234 x - 564811 \)

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:  \(-785748055861331850537130300409483=-\,13^{10}\cdot 97^{2}\cdot 347^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.13$
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:  $13, 97, 347$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{35203718507702366186015251346658355763962024264185913916012707} a^{19} - \frac{16082341106395267784051482949099533022110371758687618864862060}{35203718507702366186015251346658355763962024264185913916012707} a^{18} + \frac{566345760570613126593887222909238000214883033274092935080655}{35203718507702366186015251346658355763962024264185913916012707} a^{17} + \frac{11122134150194936489840813410765343973315798403109900691617029}{35203718507702366186015251346658355763962024264185913916012707} a^{16} + \frac{13010653693874425515511607517233514602336252256415873081322712}{35203718507702366186015251346658355763962024264185913916012707} a^{15} + \frac{8523055265599039233576089106544767504589688394137134172664017}{35203718507702366186015251346658355763962024264185913916012707} a^{14} + \frac{6840346202345046199126874602568935929498355993232230224292056}{35203718507702366186015251346658355763962024264185913916012707} a^{13} + \frac{15711012236442952684511625032725456418572859757193439458582960}{35203718507702366186015251346658355763962024264185913916012707} a^{12} + \frac{5078150534118253170491747235081864172840080845136832476544886}{35203718507702366186015251346658355763962024264185913916012707} a^{11} + \frac{11624699896657449690719348225674871164899336326399230280437297}{35203718507702366186015251346658355763962024264185913916012707} a^{10} + \frac{13292327719043337433694455068373944411048585570138187026384566}{35203718507702366186015251346658355763962024264185913916012707} a^{9} + \frac{6573535805716401234337560424058347907042449280875386657674580}{35203718507702366186015251346658355763962024264185913916012707} a^{8} - \frac{14908238105790747247239806185708654984003244074626103719952403}{35203718507702366186015251346658355763962024264185913916012707} a^{7} + \frac{14217541732192438619612794681945251245893088959016377039615765}{35203718507702366186015251346658355763962024264185913916012707} a^{6} + \frac{11707268990535347171529125943468739938654493129319273374972826}{35203718507702366186015251346658355763962024264185913916012707} a^{5} + \frac{4104344190088302127649864069140412144356343124407985206304268}{35203718507702366186015251346658355763962024264185913916012707} a^{4} - \frac{3960176274134768278323387866236551739737537360895718775318129}{35203718507702366186015251346658355763962024264185913916012707} a^{3} + \frac{14869993673714370991923986028894435526207159976363295651311771}{35203718507702366186015251346658355763962024264185913916012707} a^{2} + \frac{2183130054772647797303357013293891667079128076178106150022408}{35203718507702366186015251346658355763962024264185913916012707} a - \frac{15170623766292473790496444854117827220127728993955117936998177}{35203718507702366186015251346658355763962024264185913916012707}$

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:  \( 419170069.03 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T796:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n796 are not computed
Character table for t20n796 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 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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
347Data not computed