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Properties

Label 20.4.30694870289...3125.1
Degree $20$
Signature $[4, 8]$
Discriminant $3^{10}\cdot 5^{11}\cdot 239^{8}$
Root discriminant $37.53$
Ramified primes $3, 5, 239$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T144

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![361, -228, -1297, 1207, -2025, -416, 11721, -8718, -5714, 9385, -5899, 2527, -7, -755, 779, -450, 227, -80, 25, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 25*x^18 - 80*x^17 + 227*x^16 - 450*x^15 + 779*x^14 - 755*x^13 - 7*x^12 + 2527*x^11 - 5899*x^10 + 9385*x^9 - 5714*x^8 - 8718*x^7 + 11721*x^6 - 416*x^5 - 2025*x^4 + 1207*x^3 - 1297*x^2 - 228*x + 361)
 
gp: K = bnfinit(x^20 - 5*x^19 + 25*x^18 - 80*x^17 + 227*x^16 - 450*x^15 + 779*x^14 - 755*x^13 - 7*x^12 + 2527*x^11 - 5899*x^10 + 9385*x^9 - 5714*x^8 - 8718*x^7 + 11721*x^6 - 416*x^5 - 2025*x^4 + 1207*x^3 - 1297*x^2 - 228*x + 361, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 25 x^{18} - 80 x^{17} + 227 x^{16} - 450 x^{15} + 779 x^{14} - 755 x^{13} - 7 x^{12} + 2527 x^{11} - 5899 x^{10} + 9385 x^{9} - 5714 x^{8} - 8718 x^{7} + 11721 x^{6} - 416 x^{5} - 2025 x^{4} + 1207 x^{3} - 1297 x^{2} - 228 x + 361 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(30694870289571932375789501953125=3^{10}\cdot 5^{11}\cdot 239^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.53$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{57} a^{18} + \frac{2}{19} a^{16} + \frac{7}{57} a^{15} - \frac{23}{57} a^{14} - \frac{14}{57} a^{13} + \frac{2}{19} a^{12} - \frac{1}{19} a^{11} - \frac{22}{57} a^{10} + \frac{23}{57} a^{9} - \frac{8}{57} a^{8} + \frac{16}{57} a^{7} - \frac{10}{57} a^{6} - \frac{3}{19} a^{5} + \frac{10}{57} a^{4} - \frac{8}{19} a^{3} + \frac{2}{57} a^{2} + \frac{20}{57} a + \frac{1}{3}$, $\frac{1}{150587386069343643846959868151713139683} a^{19} + \frac{914215679195705767959205496110120820}{150587386069343643846959868151713139683} a^{18} + \frac{10041921877218382596652151770882833835}{50195795356447881282319956050571046561} a^{17} + \frac{982599478401238139627736521009770540}{150587386069343643846959868151713139683} a^{16} + \frac{2738628548230594165248135547138481800}{50195795356447881282319956050571046561} a^{15} - \frac{23621644995585607539346547361973104303}{50195795356447881282319956050571046561} a^{14} + \frac{44331281966881795061957397309082275245}{150587386069343643846959868151713139683} a^{13} + \frac{15666656621902924299063276974935489180}{50195795356447881282319956050571046561} a^{12} - \frac{46288356158322290169900988184722964173}{150587386069343643846959868151713139683} a^{11} - \frac{7804079531134875374670383641565502178}{50195795356447881282319956050571046561} a^{10} - \frac{1231261452481928283128960333504419438}{150587386069343643846959868151713139683} a^{9} + \frac{14707528340288823760208934610837499093}{50195795356447881282319956050571046561} a^{8} + \frac{62976159819791566080614246432560227553}{150587386069343643846959868151713139683} a^{7} + \frac{44629492492255591544117272786050358009}{150587386069343643846959868151713139683} a^{6} + \frac{20364929524931781912711991701046064104}{150587386069343643846959868151713139683} a^{5} - \frac{994677693396552614163550586858981276}{13689762369940331258814533468337558153} a^{4} - \frac{70282140916140153649651932900526219489}{150587386069343643846959868151713139683} a^{3} - \frac{15498789263983153744308870641340060781}{50195795356447881282319956050571046561} a^{2} + \frac{32077902993343349702285779970771833706}{150587386069343643846959868151713139683} a + \frac{3568489745844109362446489512395650696}{7925651898386507570892624639563849457}$

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:  $11$
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:  \( 102008794.555 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T144:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n144
Character table for t20n144 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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 $20$ R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
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}$
239Data not computed