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Properties

Label 20.0.73308345175...2224.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{50}\cdot 31^{4}\cdot 227^{8}$
Root discriminant $98.46$
Ramified primes $2, 31, 227$
Class number $119280$ (GRH)
Class group $[2, 2, 29820]$ (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31457276, 53328976, 91010256, 108321624, 100842924, 72089680, 46631592, 24260480, 11402270, 4473144, 1772298, 527812, 203669, 37728, 18452, 520, 1402, -96, 62, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 62*x^18 - 96*x^17 + 1402*x^16 + 520*x^15 + 18452*x^14 + 37728*x^13 + 203669*x^12 + 527812*x^11 + 1772298*x^10 + 4473144*x^9 + 11402270*x^8 + 24260480*x^7 + 46631592*x^6 + 72089680*x^5 + 100842924*x^4 + 108321624*x^3 + 91010256*x^2 + 53328976*x + 31457276)
 
gp: K = bnfinit(x^20 - 4*x^19 + 62*x^18 - 96*x^17 + 1402*x^16 + 520*x^15 + 18452*x^14 + 37728*x^13 + 203669*x^12 + 527812*x^11 + 1772298*x^10 + 4473144*x^9 + 11402270*x^8 + 24260480*x^7 + 46631592*x^6 + 72089680*x^5 + 100842924*x^4 + 108321624*x^3 + 91010256*x^2 + 53328976*x + 31457276, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 62 x^{18} - 96 x^{17} + 1402 x^{16} + 520 x^{15} + 18452 x^{14} + 37728 x^{13} + 203669 x^{12} + 527812 x^{11} + 1772298 x^{10} + 4473144 x^{9} + 11402270 x^{8} + 24260480 x^{7} + 46631592 x^{6} + 72089680 x^{5} + 100842924 x^{4} + 108321624 x^{3} + 91010256 x^{2} + 53328976 x + 31457276 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7330834517590754249461342942057121972224=2^{50}\cdot 31^{4}\cdot 227^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.46$
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:  $2, 31, 227$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{10}$, $\frac{1}{91000294145611690070125923220791460323676911679918150762488056239577895548} a^{19} - \frac{8397285194186013931025364265710605906094665816470545455919790239396585731}{45500147072805845035062961610395730161838455839959075381244028119788947774} a^{18} + \frac{9140075355627571865362762022492367513161956366607027843032333704937404301}{45500147072805845035062961610395730161838455839959075381244028119788947774} a^{17} - \frac{746846906922137522533920766151700289028216079371195102164207334994118596}{22750073536402922517531480805197865080919227919979537690622014059894473887} a^{16} - \frac{4749773498911453363229496872012854752289458486493017355810918623035650719}{45500147072805845035062961610395730161838455839959075381244028119788947774} a^{15} - \frac{2538880912315307850587303814336853339413849634441154939900450578742906721}{45500147072805845035062961610395730161838455839959075381244028119788947774} a^{14} - \frac{11375042198527053241045726183243974954964749746718963780222636356000011575}{45500147072805845035062961610395730161838455839959075381244028119788947774} a^{13} + \frac{9470617746940758644997106587275235293306821985964134227238566673826616541}{22750073536402922517531480805197865080919227919979537690622014059894473887} a^{12} - \frac{27601037191532892064559600076321803596951817486785671013372239437939776067}{91000294145611690070125923220791460323676911679918150762488056239577895548} a^{11} + \frac{2384915392299399542663035772512143707450992405822670171544918325385924525}{45500147072805845035062961610395730161838455839959075381244028119788947774} a^{10} + \frac{22314927693970825705015459603175574612020221697757723471881016499324439301}{45500147072805845035062961610395730161838455839959075381244028119788947774} a^{9} - \frac{8283226950529300970955195450916551556632532513524032168124878129958900345}{22750073536402922517531480805197865080919227919979537690622014059894473887} a^{8} + \frac{6107459709564915173034115885032303175849154961394778631732120741941983513}{45500147072805845035062961610395730161838455839959075381244028119788947774} a^{7} - \frac{21395294247937913061557171569853527589375854975670173677975935324151781337}{45500147072805845035062961610395730161838455839959075381244028119788947774} a^{6} + \frac{11128528028990503991871289180771697837401141797163644205085351545552493297}{45500147072805845035062961610395730161838455839959075381244028119788947774} a^{5} + \frac{4040142132396902346448752094371788346540594889742486989344131334513536441}{22750073536402922517531480805197865080919227919979537690622014059894473887} a^{4} + \frac{5329188831908703897110261303359675997077109505176375320081613560439102433}{22750073536402922517531480805197865080919227919979537690622014059894473887} a^{3} - \frac{2146395868453306814323696364365839875911102109474801522166904329468813266}{22750073536402922517531480805197865080919227919979537690622014059894473887} a^{2} + \frac{4239854026537414811937193115023313109527977975670230513533346300003205389}{22750073536402922517531480805197865080919227919979537690622014059894473887} a + \frac{5366176115915366493556209590760051798823788261093221439773331084651403316}{22750073536402922517531480805197865080919227919979537690622014059894473887}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{29820}$, which has order $119280$ (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:  $9$
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:  \( 10717188.4751 \) (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

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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 R $16{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ R $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.24.10$x^{8} + 16$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.12.26.27$x^{12} - 2 x^{10} + 4 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{3} + 2$$12$$1$$26$12T48$[4/3, 4/3, 2, 3]_{3}^{2}$
31Data not computed
227Data not computed