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Properties

Label 20.8.98068750829...0625.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{16}\cdot 6329^{6}$
Root discriminant $50.07$
Ramified primes $5, 6329$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-120475, 594300, -1007000, 517250, 314185, -330770, -47925, 47430, 50641, -20912, 2434, -3556, 1478, 64, -355, 116, 86, -42, -29, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 29*x^18 - 42*x^17 + 86*x^16 + 116*x^15 - 355*x^14 + 64*x^13 + 1478*x^12 - 3556*x^11 + 2434*x^10 - 20912*x^9 + 50641*x^8 + 47430*x^7 - 47925*x^6 - 330770*x^5 + 314185*x^4 + 517250*x^3 - 1007000*x^2 + 594300*x - 120475)
 
gp: K = bnfinit(x^20 - 29*x^18 - 42*x^17 + 86*x^16 + 116*x^15 - 355*x^14 + 64*x^13 + 1478*x^12 - 3556*x^11 + 2434*x^10 - 20912*x^9 + 50641*x^8 + 47430*x^7 - 47925*x^6 - 330770*x^5 + 314185*x^4 + 517250*x^3 - 1007000*x^2 + 594300*x - 120475, 1)
 

Normalized defining polynomial

\( x^{20} - 29 x^{18} - 42 x^{17} + 86 x^{16} + 116 x^{15} - 355 x^{14} + 64 x^{13} + 1478 x^{12} - 3556 x^{11} + 2434 x^{10} - 20912 x^{9} + 50641 x^{8} + 47430 x^{7} - 47925 x^{6} - 330770 x^{5} + 314185 x^{4} + 517250 x^{3} - 1007000 x^{2} + 594300 x - 120475 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9806875082992983948596343994140625=5^{16}\cdot 6329^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.07$
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:  $5, 6329$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{10} a^{16} + \frac{1}{10} a^{14} - \frac{1}{5} a^{13} + \frac{1}{10} a^{12} + \frac{1}{10} a^{11} - \frac{1}{10} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{10} a^{6} + \frac{3}{10} a^{5} + \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{17} + \frac{1}{10} a^{15} - \frac{1}{5} a^{14} + \frac{1}{10} a^{13} + \frac{1}{10} a^{12} - \frac{1}{10} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{10} a^{7} + \frac{3}{10} a^{6} + \frac{1}{10} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{20} a^{18} - \frac{1}{20} a^{17} - \frac{1}{20} a^{16} + \frac{1}{10} a^{15} + \frac{1}{20} a^{14} - \frac{1}{20} a^{13} - \frac{3}{20} a^{12} + \frac{1}{10} a^{11} - \frac{1}{20} a^{10} - \frac{7}{20} a^{9} + \frac{1}{5} a^{8} + \frac{1}{20} a^{7} + \frac{3}{20} a^{5} + \frac{2}{5} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{641122861321147362121808666529390720583839931834763780} a^{19} - \frac{1743264719439908521387602834232540003247314573514493}{128224572264229472424361733305878144116767986366952756} a^{18} + \frac{30642008567728117114980510603669348778656096978469333}{641122861321147362121808666529390720583839931834763780} a^{17} + \frac{2180876126592035611281315779950528576530315164181063}{320561430660573681060904333264695360291919965917381890} a^{16} + \frac{138461482103847709984953311637650426141963078947192323}{641122861321147362121808666529390720583839931834763780} a^{15} - \frac{21681466561002017905089128596933177413289334047757793}{128224572264229472424361733305878144116767986366952756} a^{14} + \frac{72769654727749610118048733542403493139023914525672961}{641122861321147362121808666529390720583839931834763780} a^{13} + \frac{25668375262240087782460813096670794934864960994621067}{320561430660573681060904333264695360291919965917381890} a^{12} - \frac{95177416928103993707756340636176780161120810165757439}{641122861321147362121808666529390720583839931834763780} a^{11} - \frac{117485420888276623175883204507342768407805320397593743}{641122861321147362121808666529390720583839931834763780} a^{10} + \frac{8896359844582440019668850066834330185740383215737741}{320561430660573681060904333264695360291919965917381890} a^{9} + \frac{59607187709481871516905117141542281365745941688147331}{128224572264229472424361733305878144116767986366952756} a^{8} + \frac{106725492240335083549313216056402668986150816860636933}{320561430660573681060904333264695360291919965917381890} a^{7} + \frac{223420721160792010780021845222588502522735786008364523}{641122861321147362121808666529390720583839931834763780} a^{6} - \frac{75659178644542662378946332041591089964768854572736857}{320561430660573681060904333264695360291919965917381890} a^{5} - \frac{177366608084078671131535683945077775524576754040994587}{641122861321147362121808666529390720583839931834763780} a^{4} + \frac{426854722901161089421116680422675087992002970797675}{2210768487314301248695891953549623174427034247706082} a^{3} + \frac{18875524687283422566550642517330097490061604543941085}{128224572264229472424361733305878144116767986366952756} a^{2} + \frac{5984547154012268517092642680278702447643098170979707}{128224572264229472424361733305878144116767986366952756} a - \frac{267230973606894620152169990524725186225012726291229}{811547925722971344457985653834671798207392318778182}$

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:  $13$
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:  \( 5121401083.12 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.625878765625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ $16{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.8.0.1}{8} }$ $16{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
6329Data not computed