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Properties

Label 20.0.24113403175...5625.2
Degree $20$
Signature $[0, 10]$
Discriminant $5^{34}\cdot 23^{10}$
Root discriminant $73.98$
Ramified primes $5, 23$
Class number $112053$ (GRH)
Class group $[112053]$ (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![795562849, -617763370, 878529790, -560610915, 454340820, -242935385, 145105405, -66095535, 31777520, -12451945, 4996534, -1685235, 571270, -164095, 46735, -11130, 2590, -480, 85, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 85*x^18 - 480*x^17 + 2590*x^16 - 11130*x^15 + 46735*x^14 - 164095*x^13 + 571270*x^12 - 1685235*x^11 + 4996534*x^10 - 12451945*x^9 + 31777520*x^8 - 66095535*x^7 + 145105405*x^6 - 242935385*x^5 + 454340820*x^4 - 560610915*x^3 + 878529790*x^2 - 617763370*x + 795562849)
 
gp: K = bnfinit(x^20 - 10*x^19 + 85*x^18 - 480*x^17 + 2590*x^16 - 11130*x^15 + 46735*x^14 - 164095*x^13 + 571270*x^12 - 1685235*x^11 + 4996534*x^10 - 12451945*x^9 + 31777520*x^8 - 66095535*x^7 + 145105405*x^6 - 242935385*x^5 + 454340820*x^4 - 560610915*x^3 + 878529790*x^2 - 617763370*x + 795562849, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 85 x^{18} - 480 x^{17} + 2590 x^{16} - 11130 x^{15} + 46735 x^{14} - 164095 x^{13} + 571270 x^{12} - 1685235 x^{11} + 4996534 x^{10} - 12451945 x^{9} + 31777520 x^{8} - 66095535 x^{7} + 145105405 x^{6} - 242935385 x^{5} + 454340820 x^{4} - 560610915 x^{3} + 878529790 x^{2} - 617763370 x + 795562849 \)

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:  \(24113403175520361401140689849853515625=5^{34}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.98$
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:  $5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(575=5^{2}\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{575}(1,·)$, $\chi_{575}(139,·)$, $\chi_{575}(321,·)$, $\chi_{575}(459,·)$, $\chi_{575}(461,·)$, $\chi_{575}(206,·)$, $\chi_{575}(24,·)$, $\chi_{575}(344,·)$, $\chi_{575}(346,·)$, $\chi_{575}(91,·)$, $\chi_{575}(484,·)$, $\chi_{575}(229,·)$, $\chi_{575}(551,·)$, $\chi_{575}(231,·)$, $\chi_{575}(369,·)$, $\chi_{575}(114,·)$, $\chi_{575}(116,·)$, $\chi_{575}(254,·)$, $\chi_{575}(436,·)$, $\chi_{575}(574,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{7} a^{17} + \frac{3}{7} a^{16} - \frac{2}{7} a^{15} + \frac{3}{7} a^{14} - \frac{2}{7} a^{13} - \frac{1}{7} a^{12} - \frac{2}{7} a^{11} - \frac{2}{7} a^{10} - \frac{3}{7} a^{9} + \frac{1}{7} a^{8} - \frac{3}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{3}{7} a^{3} - \frac{2}{7} a^{2} + \frac{2}{7}$, $\frac{1}{7} a^{18} + \frac{3}{7} a^{16} + \frac{2}{7} a^{15} + \frac{3}{7} a^{14} - \frac{2}{7} a^{13} + \frac{1}{7} a^{12} - \frac{3}{7} a^{11} + \frac{3}{7} a^{10} + \frac{3}{7} a^{9} - \frac{3}{7} a^{8} - \frac{3}{7} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{1}{7} a^{2} + \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{626680933233457107588108567090522517663269742044188991006504757} a^{19} + \frac{914386474963287950911058132894266048055687828393535668172539}{14573975191475746688095548071872616689843482373120674209453599} a^{18} - \frac{22184573114812850991036753899713992594613678108415277759104991}{626680933233457107588108567090522517663269742044188991006504757} a^{17} + \frac{154949490425964392108925650233687834236238975984061072793045363}{626680933233457107588108567090522517663269742044188991006504757} a^{16} - \frac{37512766210030310248238336123161174042844773950947028055210262}{89525847604779586798301223870074645380467106006312713000929251} a^{15} + \frac{13725484866595696131800410179560566700292190681064088244576533}{626680933233457107588108567090522517663269742044188991006504757} a^{14} - \frac{113490473318278703659914852014220120551559777294751952542800596}{626680933233457107588108567090522517663269742044188991006504757} a^{13} - \frac{211421451775828243665406621455508600306177020741781756800332990}{626680933233457107588108567090522517663269742044188991006504757} a^{12} - \frac{161569415193658717788692957959882588351019006715883735159816455}{626680933233457107588108567090522517663269742044188991006504757} a^{11} + \frac{247738708693147511759144508420939976470674730872448507483759810}{626680933233457107588108567090522517663269742044188991006504757} a^{10} - \frac{75786148409112786278305064880736473884907036708242273344792380}{626680933233457107588108567090522517663269742044188991006504757} a^{9} - \frac{295365743829644815130182803679715353675332482111795999238882236}{626680933233457107588108567090522517663269742044188991006504757} a^{8} + \frac{273781751800575164230010952720451894427434531991318611215395051}{626680933233457107588108567090522517663269742044188991006504757} a^{7} + \frac{25716017613572930746232358279749968751440571464757040359190487}{89525847604779586798301223870074645380467106006312713000929251} a^{6} + \frac{4820360740663505016719756490554767068672584322070111979266846}{14573975191475746688095548071872616689843482373120674209453599} a^{5} - \frac{152661142256400742351973698873538301652088076055662622792172863}{626680933233457107588108567090522517663269742044188991006504757} a^{4} + \frac{10161340139521122772908070180945103134413065521559380036720642}{89525847604779586798301223870074645380467106006312713000929251} a^{3} - \frac{8461210939768149807514228087660152043144317347111969245598780}{626680933233457107588108567090522517663269742044188991006504757} a^{2} - \frac{247777498889584688973283522301105241453831963238534545910636268}{626680933233457107588108567090522517663269742044188991006504757} a - \frac{91762409306501322537579107376704608868711000410799751024012698}{626680933233457107588108567090522517663269742044188991006504757}$

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

Class group and class number

$C_{112053}$, which has order $112053$ (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:  \( 161406.837641 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-115}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{5}, \sqrt{-23})\), 5.5.390625.1, \(\Q(\zeta_{25})^+\), 10.0.4910540008544921875.3, 10.0.982108001708984375.4

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
5Data not computed
23Data not computed