↑

Properties

Label 20.2.78574805586...9483.1
Degree $20$
Signature $[2, 9]$
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

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1444031, -500979, 6073672, -11186429, 10789583, -8938579, 4299577, -2208621, 354518, -99930, -116322, 36977, -27523, 4757, -1125, -292, 274, -86, 33, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 33*x^18 - 86*x^17 + 274*x^16 - 292*x^15 - 1125*x^14 + 4757*x^13 - 27523*x^12 + 36977*x^11 - 116322*x^10 - 99930*x^9 + 354518*x^8 - 2208621*x^7 + 4299577*x^6 - 8938579*x^5 + 10789583*x^4 - 11186429*x^3 + 6073672*x^2 - 500979*x - 1444031)
 
gp: K = bnfinit(x^20 - 4*x^19 + 33*x^18 - 86*x^17 + 274*x^16 - 292*x^15 - 1125*x^14 + 4757*x^13 - 27523*x^12 + 36977*x^11 - 116322*x^10 - 99930*x^9 + 354518*x^8 - 2208621*x^7 + 4299577*x^6 - 8938579*x^5 + 10789583*x^4 - 11186429*x^3 + 6073672*x^2 - 500979*x - 1444031, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 33 x^{18} - 86 x^{17} + 274 x^{16} - 292 x^{15} - 1125 x^{14} + 4757 x^{13} - 27523 x^{12} + 36977 x^{11} - 116322 x^{10} - 99930 x^{9} + 354518 x^{8} - 2208621 x^{7} + 4299577 x^{6} - 8938579 x^{5} + 10789583 x^{4} - 11186429 x^{3} + 6073672 x^{2} - 500979 x - 1444031 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{13} a^{18} + \frac{2}{13} a^{17} - \frac{3}{13} a^{16} - \frac{5}{13} a^{15} - \frac{2}{13} a^{14} + \frac{1}{13} a^{13} + \frac{4}{13} a^{12} + \frac{1}{13} a^{11} - \frac{6}{13} a^{10} - \frac{1}{13} a^{9} - \frac{2}{13} a^{8} - \frac{2}{13} a^{7} + \frac{1}{13} a^{6} - \frac{1}{13} a^{5} - \frac{6}{13} a^{4} - \frac{1}{13} a^{3} - \frac{5}{13} a^{2} - \frac{2}{13} a + \frac{1}{13}$, $\frac{1}{305000423569926775186348749484427734381180436894597185289370353} a^{19} + \frac{283197510341195211779212109648420286387464930173519604544846}{23461571043840521168180673037263671875475418222661321945336181} a^{18} - \frac{130148665602479273663020732943227289052485452083095419618337429}{305000423569926775186348749484427734381180436894597185289370353} a^{17} - \frac{39552831571183842582890227041426565290998198343607335328427708}{305000423569926775186348749484427734381180436894597185289370353} a^{16} + \frac{121795410981996471073837058111791127921432612406923481643990208}{305000423569926775186348749484427734381180436894597185289370353} a^{15} + \frac{119505008846962928156778715714283946533743453836474373962400719}{305000423569926775186348749484427734381180436894597185289370353} a^{14} - \frac{19934657445884511368056784414011069684160576660865441715168113}{305000423569926775186348749484427734381180436894597185289370353} a^{13} - \frac{151551885335287747288982443746494866493774917637155601970091123}{305000423569926775186348749484427734381180436894597185289370353} a^{12} + \frac{2575297406221034398953135927660448450600306124858093936242820}{5754724973017486324270731122347693101531706356501833684705101} a^{11} + \frac{89843637390320027523366944725856971337124876444491505212190130}{305000423569926775186348749484427734381180436894597185289370353} a^{10} + \frac{11367054438269994391914299733842134435540717196240134585627046}{23461571043840521168180673037263671875475418222661321945336181} a^{9} + \frac{92610622483668244176809521724604829973099893321258295106263351}{305000423569926775186348749484427734381180436894597185289370353} a^{8} - \frac{30056631997902118295484625210455463724245533860600852683785159}{305000423569926775186348749484427734381180436894597185289370353} a^{7} - \frac{61259412902973130051450633608990393498739339972181405831365595}{305000423569926775186348749484427734381180436894597185289370353} a^{6} + \frac{69335788815230184404682889909392256290021295032473720450920525}{305000423569926775186348749484427734381180436894597185289370353} a^{5} - \frac{97531608430612044969479140533691783635028112061221623616979429}{305000423569926775186348749484427734381180436894597185289370353} a^{4} + \frac{24603156543872680152189354054304071596072712634026360467978700}{305000423569926775186348749484427734381180436894597185289370353} a^{3} - \frac{27007540362834423210567878517604742704443406528945034207418199}{305000423569926775186348749484427734381180436894597185289370353} a^{2} - \frac{9989051125316559476853619780236910102852633328001583622245619}{305000423569926775186348749484427734381180436894597185289370353} a + \frac{32447472559178287924332824014963882214098124322647113282529052}{305000423569926775186348749484427734381180436894597185289370353}$

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:  $10$
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:  \( 169042557.959 \) (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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$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}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
347Data not computed