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Properties

Label 20.0.35291773913...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 5^{32}\cdot 13^{10}$
Root discriminant $189.40$
Ramified primes $2, 5, 13$
Class number $79705650$ (GRH)
Class group $[5, 15941130]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11102390257, 2623654180, 6743617240, 1276163740, 1892348205, 289047932, 329003610, 38866760, 39836375, 3347020, 3602088, 194160, 253690, 7040, 14390, 116, 645, -20, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 20*x^18 - 20*x^17 + 645*x^16 + 116*x^15 + 14390*x^14 + 7040*x^13 + 253690*x^12 + 194160*x^11 + 3602088*x^10 + 3347020*x^9 + 39836375*x^8 + 38866760*x^7 + 329003610*x^6 + 289047932*x^5 + 1892348205*x^4 + 1276163740*x^3 + 6743617240*x^2 + 2623654180*x + 11102390257)
 
gp: K = bnfinit(x^20 + 20*x^18 - 20*x^17 + 645*x^16 + 116*x^15 + 14390*x^14 + 7040*x^13 + 253690*x^12 + 194160*x^11 + 3602088*x^10 + 3347020*x^9 + 39836375*x^8 + 38866760*x^7 + 329003610*x^6 + 289047932*x^5 + 1892348205*x^4 + 1276163740*x^3 + 6743617240*x^2 + 2623654180*x + 11102390257, 1)
 

Normalized defining polynomial

\( x^{20} + 20 x^{18} - 20 x^{17} + 645 x^{16} + 116 x^{15} + 14390 x^{14} + 7040 x^{13} + 253690 x^{12} + 194160 x^{11} + 3602088 x^{10} + 3347020 x^{9} + 39836375 x^{8} + 38866760 x^{7} + 329003610 x^{6} + 289047932 x^{5} + 1892348205 x^{4} + 1276163740 x^{3} + 6743617240 x^{2} + 2623654180 x + 11102390257 \)

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:  \(3529177391334400000000000000000000000000000000=2^{40}\cdot 5^{32}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $189.40$
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:  $2, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2600=2^{3}\cdot 5^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2600}(1,·)$, $\chi_{2600}(1091,·)$, $\chi_{2600}(261,·)$, $\chi_{2600}(1351,·)$, $\chi_{2600}(521,·)$, $\chi_{2600}(1611,·)$, $\chi_{2600}(781,·)$, $\chi_{2600}(1871,·)$, $\chi_{2600}(1041,·)$, $\chi_{2600}(2131,·)$, $\chi_{2600}(1301,·)$, $\chi_{2600}(2391,·)$, $\chi_{2600}(1561,·)$, $\chi_{2600}(1821,·)$, $\chi_{2600}(2081,·)$, $\chi_{2600}(2341,·)$, $\chi_{2600}(51,·)$, $\chi_{2600}(311,·)$, $\chi_{2600}(571,·)$, $\chi_{2600}(831,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{7} a^{11} - \frac{1}{7} a^{5}$, $\frac{1}{7} a^{12} - \frac{1}{7} a^{6}$, $\frac{1}{49} a^{13} + \frac{3}{49} a^{12} + \frac{2}{49} a^{11} - \frac{1}{49} a^{10} - \frac{3}{49} a^{9} - \frac{2}{49} a^{8} - \frac{1}{49} a^{7} + \frac{18}{49} a^{6} + \frac{12}{49} a^{5} - \frac{6}{49} a^{4} - \frac{18}{49} a^{3} - \frac{12}{49} a^{2} + \frac{1}{7} a$, $\frac{1}{49} a^{14} - \frac{2}{49} a^{8} + \frac{1}{49} a^{2}$, $\frac{1}{49} a^{15} - \frac{2}{49} a^{9} + \frac{1}{49} a^{3}$, $\frac{1}{49} a^{16} - \frac{2}{49} a^{10} + \frac{1}{49} a^{4}$, $\frac{1}{343} a^{17} - \frac{1}{343} a^{15} + \frac{2}{343} a^{14} + \frac{1}{343} a^{13} - \frac{4}{343} a^{12} + \frac{2}{49} a^{11} + \frac{20}{343} a^{10} + \frac{13}{343} a^{9} + \frac{15}{343} a^{8} - \frac{1}{343} a^{7} - \frac{171}{343} a^{6} - \frac{99}{343} a^{5} + \frac{22}{343} a^{4} + \frac{163}{343} a^{3} + \frac{165}{343} a^{2} + \frac{8}{49} a + \frac{2}{7}$, $\frac{1}{1645503364949143830907} a^{18} - \frac{1118256242435930823}{1645503364949143830907} a^{17} + \frac{2119872142865311883}{1645503364949143830907} a^{16} + \frac{3406769324082235121}{1645503364949143830907} a^{15} - \frac{10223424912187942029}{1645503364949143830907} a^{14} + \frac{9488563401656686881}{1645503364949143830907} a^{13} + \frac{92938501658896673217}{1645503364949143830907} a^{12} - \frac{44454953856104774277}{1645503364949143830907} a^{11} - \frac{16637778848400070623}{235071909278449118701} a^{10} - \frac{19438504150846013868}{1645503364949143830907} a^{9} - \frac{5477435624308721908}{1645503364949143830907} a^{8} + \frac{9319114634230453057}{1645503364949143830907} a^{7} - \frac{595960398123800277286}{1645503364949143830907} a^{6} - \frac{237042949015926813478}{1645503364949143830907} a^{5} - \frac{211534481774191881962}{1645503364949143830907} a^{4} - \frac{796172408076845913811}{1645503364949143830907} a^{3} + \frac{493318652240624143895}{1645503364949143830907} a^{2} + \frac{11071269238394914072}{235071909278449118701} a + \frac{11005257822718739312}{33581701325492731243}$, $\frac{1}{198369984663896754222009742580062443108214516843} a^{19} - \frac{60019127546732073701779045}{198369984663896754222009742580062443108214516843} a^{18} + \frac{240048955009371253262399251964813498876114618}{198369984663896754222009742580062443108214516843} a^{17} + \frac{876957130261453743110024766765991264813773997}{198369984663896754222009742580062443108214516843} a^{16} + \frac{167376258906421070722438021446976579809418009}{28338569237699536317429963225723206158316359549} a^{15} - \frac{997894581386531821804240502921844723437264858}{198369984663896754222009742580062443108214516843} a^{14} - \frac{193453883248594555273213827901742596031718607}{28338569237699536317429963225723206158316359549} a^{13} - \frac{4163020972460440908101505855879920605211396453}{198369984663896754222009742580062443108214516843} a^{12} + \frac{14024818129529236639491922176508436505050408906}{198369984663896754222009742580062443108214516843} a^{11} + \frac{13098090684490482862953398714084263932085984436}{198369984663896754222009742580062443108214516843} a^{10} - \frac{7613843858069669603560626307138540409997674504}{198369984663896754222009742580062443108214516843} a^{9} + \frac{10892388032112916705078393008895819153316117974}{198369984663896754222009742580062443108214516843} a^{8} - \frac{6801759486318043946789293163941229789490753930}{198369984663896754222009742580062443108214516843} a^{7} + \frac{7191835605042041796780710133417421651586106035}{198369984663896754222009742580062443108214516843} a^{6} + \frac{77761749801311790111311259694393611119927021796}{198369984663896754222009742580062443108214516843} a^{5} + \frac{792066015880070377914946741033054821220853009}{28338569237699536317429963225723206158316359549} a^{4} - \frac{81881294675795765274145006779403386939423216467}{198369984663896754222009742580062443108214516843} a^{3} + \frac{39849910942253191319050553225921906002993808560}{198369984663896754222009742580062443108214516843} a^{2} - \frac{11020364769945468377730062280445367126459518257}{28338569237699536317429963225723206158316359549} a - \frac{549984468432511443868154713970468634966536614}{4048367033957076616775709032246172308330908507}$

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

Class group and class number

$C_{5}\times C_{15941130}$, which has order $79705650$ (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:  \( 42294001.73672045 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{2}, \sqrt{-13})\), 5.5.390625.1, 10.10.5000000000000000.1, 10.0.58014531250000000000.3, 10.0.1856465000000000000000.3

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.10.16.7$x^{10} + 40 x^{9} + 10 x^{8} + 70 x^{7} + 15 x^{6} + 95 x^{5} + 5 x^{4} + 80 x^{3} + 5 x^{2} + 90 x + 7$$5$$2$$16$$C_{10}$$[2]^{2}$
5.10.16.7$x^{10} + 40 x^{9} + 10 x^{8} + 70 x^{7} + 15 x^{6} + 95 x^{5} + 5 x^{4} + 80 x^{3} + 5 x^{2} + 90 x + 7$$5$$2$$16$$C_{10}$$[2]^{2}$
13Data not computed