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Properties

Label 20.0.12106821425...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{23}\cdot 89^{16}$
Root discriminant $401.93$
Ramified primes $2, 5, 89$
Class number $655360000$ (GRH)
Class group $[4, 4, 4, 4, 40, 40, 40, 40]$ (GRH)
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16243247601, -227496465, -451806705, -442292655, -202273790, 134738047, 23133065, 9027905, -476720, -2214685, 1032069, -130305, 16340, 6865, -5155, 1303, 70, -35, 15, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 15*x^18 - 35*x^17 + 70*x^16 + 1303*x^15 - 5155*x^14 + 6865*x^13 + 16340*x^12 - 130305*x^11 + 1032069*x^10 - 2214685*x^9 - 476720*x^8 + 9027905*x^7 + 23133065*x^6 + 134738047*x^5 - 202273790*x^4 - 442292655*x^3 - 451806705*x^2 - 227496465*x + 16243247601)
 
gp: K = bnfinit(x^20 - 5*x^19 + 15*x^18 - 35*x^17 + 70*x^16 + 1303*x^15 - 5155*x^14 + 6865*x^13 + 16340*x^12 - 130305*x^11 + 1032069*x^10 - 2214685*x^9 - 476720*x^8 + 9027905*x^7 + 23133065*x^6 + 134738047*x^5 - 202273790*x^4 - 442292655*x^3 - 451806705*x^2 - 227496465*x + 16243247601, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 15 x^{18} - 35 x^{17} + 70 x^{16} + 1303 x^{15} - 5155 x^{14} + 6865 x^{13} + 16340 x^{12} - 130305 x^{11} + 1032069 x^{10} - 2214685 x^{9} - 476720 x^{8} + 9027905 x^{7} + 23133065 x^{6} + 134738047 x^{5} - 202273790 x^{4} - 442292655 x^{3} - 451806705 x^{2} - 227496465 x + 16243247601 \)

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:  \(12106821425921044089921349789500781250000000000000000=2^{16}\cdot 5^{23}\cdot 89^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $401.93$
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:  $2, 5, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} + \frac{2}{5}$, $\frac{1}{5} a^{6} + \frac{2}{5} a$, $\frac{1}{5} a^{7} + \frac{2}{5} a^{2}$, $\frac{1}{25} a^{8} + \frac{1}{25} a^{7} + \frac{2}{25} a^{6} - \frac{2}{25} a^{5} + \frac{12}{25} a^{3} + \frac{12}{25} a^{2} - \frac{1}{25} a + \frac{1}{25}$, $\frac{1}{25} a^{9} + \frac{1}{25} a^{7} + \frac{1}{25} a^{6} + \frac{2}{25} a^{5} + \frac{12}{25} a^{4} + \frac{12}{25} a^{2} + \frac{12}{25} a - \frac{1}{25}$, $\frac{1}{25} a^{10} - \frac{1}{25} a^{5} - \frac{6}{25}$, $\frac{1}{125} a^{11} + \frac{2}{125} a^{10} - \frac{1}{125} a^{6} - \frac{2}{125} a^{5} - \frac{31}{125} a - \frac{62}{125}$, $\frac{1}{250} a^{12} - \frac{1}{250} a^{11} + \frac{2}{125} a^{10} - \frac{1}{50} a^{8} - \frac{3}{125} a^{7} - \frac{9}{250} a^{6} + \frac{3}{125} a^{5} - \frac{1}{2} a^{4} - \frac{6}{25} a^{3} + \frac{17}{125} a^{2} - \frac{89}{250} a + \frac{121}{250}$, $\frac{1}{250} a^{13} - \frac{1}{250} a^{11} - \frac{2}{125} a^{10} - \frac{1}{50} a^{9} - \frac{1}{250} a^{8} - \frac{1}{50} a^{7} + \frac{21}{250} a^{6} + \frac{19}{250} a^{5} + \frac{13}{50} a^{4} + \frac{47}{125} a^{3} + \frac{13}{50} a^{2} - \frac{52}{125} a - \frac{71}{250}$, $\frac{1}{17750} a^{14} - \frac{1}{8875} a^{13} + \frac{2}{8875} a^{12} + \frac{63}{17750} a^{11} - \frac{197}{17750} a^{10} - \frac{11}{17750} a^{9} + \frac{11}{8875} a^{8} + \frac{1731}{17750} a^{7} - \frac{879}{8875} a^{6} + \frac{37}{17750} a^{5} + \frac{7099}{17750} a^{4} - \frac{5323}{17750} a^{3} - \frac{1777}{8875} a^{2} - \frac{209}{8875} a - \frac{4063}{17750}$, $\frac{1}{88750} a^{15} - \frac{689}{88750} a^{10} + \frac{3907}{88750} a^{5} - \frac{15297}{88750}$, $\frac{1}{88750} a^{16} + \frac{21}{88750} a^{11} + \frac{2}{125} a^{10} + \frac{3197}{88750} a^{6} - \frac{2}{125} a^{5} - \frac{37307}{88750} a - \frac{62}{125}$, $\frac{1}{508410923448303750} a^{17} + \frac{925879346309}{254205461724151875} a^{16} - \frac{271866161148}{84735153908050625} a^{15} - \frac{5920915481}{290520527684745} a^{14} - \frac{4299416781401}{2905205276847450} a^{13} + \frac{833130439722631}{508410923448303750} a^{12} - \frac{554235135593527}{508410923448303750} a^{11} + \frac{141784951380176}{254205461724151875} a^{10} + \frac{130171163015513}{20336436937932150} a^{9} - \frac{224169905059}{11392961869990} a^{8} - \frac{6345700575534148}{84735153908050625} a^{7} - \frac{19060374154174399}{508410923448303750} a^{6} + \frac{15081234050827939}{508410923448303750} a^{5} - \frac{4296367623168289}{20336436937932150} a^{4} - \frac{302287108975736}{10168218468966075} a^{3} - \frac{85572239884144771}{254205461724151875} a^{2} + \frac{97766214231765752}{254205461724151875} a + \frac{595392669659563}{1424120233748750}$, $\frac{1}{181502699671044438750} a^{18} - \frac{1}{36300539934208887750} a^{17} + \frac{31897594177924}{30250449945174073125} a^{16} + \frac{40973744903078}{12964478547931745625} a^{15} - \frac{136629079825453}{5185791419172698250} a^{14} + \frac{36971631357875603}{90751349835522219375} a^{13} + \frac{32446962835143421}{36300539934208887750} a^{12} + \frac{226855371562808057}{90751349835522219375} a^{11} - \frac{2910375723297227953}{181502699671044438750} a^{10} + \frac{691814872681039}{50841092344830375} a^{9} + \frac{286212858128874527}{30250449945174073125} a^{8} - \frac{647747538719946509}{36300539934208887750} a^{7} - \frac{14399780583448681847}{181502699671044438750} a^{6} + \frac{580942840695589042}{90751349835522219375} a^{5} + \frac{1354913820742415513}{18150269967104443875} a^{4} + \frac{41654925897561419504}{90751349835522219375} a^{3} + \frac{97611665396666129}{3630053993420888775} a^{2} - \frac{114926889211909582}{254205461724151875} a - \frac{64491681162833}{712060116874375}$, $\frac{1}{323982318912814323168750} a^{19} - \frac{181}{161991159456407161584375} a^{18} + \frac{4787}{11999345144919049006250} a^{17} + \frac{6611370790973534}{3305942029722595134375} a^{16} - \frac{52480722840997831}{23141594208058165940625} a^{15} + \frac{8858137852137922111}{323982318912814323168750} a^{14} - \frac{137025588739770139657}{323982318912814323168750} a^{13} - \frac{267542045490790143718}{161991159456407161584375} a^{12} - \frac{135352434768061429274}{161991159456407161584375} a^{11} - \frac{3410568996075159266}{453756749177611096875} a^{10} + \frac{563408214105382493747}{53997053152135720528125} a^{9} + \frac{2503191141092420543308}{161991159456407161584375} a^{8} + \frac{2271368559332372486984}{161991159456407161584375} a^{7} + \frac{1524643542838568030962}{161991159456407161584375} a^{6} - \frac{1362799941100284714163}{323982318912814323168750} a^{5} - \frac{3312383933262397985597}{323982318912814323168750} a^{4} - \frac{5567486694937386206518}{161991159456407161584375} a^{3} - \frac{22487260926887132553}{50417416575290121875} a^{2} + \frac{404492975322623479}{2542054617241518750} a - \frac{742790117688982}{3560300584371875}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{4}\times C_{4}\times C_{40}\times C_{40}\times C_{40}\times C_{40}$, which has order $655360000$ (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:  \( \frac{2607775861192}{32398231891281432316875} a^{19} - \frac{6380649266219}{32398231891281432316875} a^{18} + \frac{6380649266219}{10799410630427144105625} a^{17} - \frac{6380649266219}{4628318841611633188125} a^{16} + \frac{12761298532438}{4628318841611633188125} a^{15} + \frac{3656650345484827}{32398231891281432316875} a^{14} - \frac{6578449393471789}{32398231891281432316875} a^{13} + \frac{8760631442518687}{32398231891281432316875} a^{12} + \frac{20851961802003692}{32398231891281432316875} a^{11} - \frac{465787396433987}{90751349835522219375} a^{10} + \frac{729607852605333343}{10799410630427144105625} a^{9} - \frac{2826225644031245203}{32398231891281432316875} a^{8} - \frac{608356623638384336}{32398231891281432316875} a^{7} + \frac{11520779082748968239}{32398231891281432316875} a^{6} + \frac{29520794843529286247}{32398231891281432316875} a^{5} + \frac{924649262165373325036}{32398231891281432316875} a^{4} - \frac{258127621947767220002}{32398231891281432316875} a^{3} - \frac{1581016417131542477}{90751349835522219375} a^{2} - \frac{4523880329749271}{254205461724151875} a - \frac{6380649266219}{712060116874375} \) (order $10$)
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:  \( 3260665403.950658 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.3137112050000.2 x5, 10.2.49207360071276012500000000.1 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 5 sibling: data not computed
Degree 10 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
2Data not computed
5Data not computed
$89$89.10.8.1$x^{10} - 13439 x^{5} + 61593696$$5$$2$$8$$D_5$$[\ ]_{5}^{2}$
89.10.8.1$x^{10} - 13439 x^{5} + 61593696$$5$$2$$8$$D_5$$[\ ]_{5}^{2}$