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Properties

Label 20.0.44322362984...0000.6
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 31^{18}$
Root discriminant $340.67$
Ramified primes $2, 3, 5, 31$
Class number $412972032$ (GRH)
Class group $[2, 2, 2, 2, 2, 22, 44, 13332]$ (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![312893678161, 0, 171331481608, 0, 58165466967, 0, 8331032808, 0, 476229607, 0, -19145372, 0, 2159792, 0, -114312, 0, 1397, 0, -32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 32*x^18 + 1397*x^16 - 114312*x^14 + 2159792*x^12 - 19145372*x^10 + 476229607*x^8 + 8331032808*x^6 + 58165466967*x^4 + 171331481608*x^2 + 312893678161)
 
gp: K = bnfinit(x^20 - 32*x^18 + 1397*x^16 - 114312*x^14 + 2159792*x^12 - 19145372*x^10 + 476229607*x^8 + 8331032808*x^6 + 58165466967*x^4 + 171331481608*x^2 + 312893678161, 1)
 

Normalized defining polynomial

\( x^{20} - 32 x^{18} + 1397 x^{16} - 114312 x^{14} + 2159792 x^{12} - 19145372 x^{10} + 476229607 x^{8} + 8331032808 x^{6} + 58165466967 x^{4} + 171331481608 x^{2} + 312893678161 \)

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:  \(443223629840246396758117587996045905756160000000000=2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $340.67$
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:  $2, 3, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3720=2^{3}\cdot 3\cdot 5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{3720}(1,·)$, $\chi_{3720}(3719,·)$, $\chi_{3720}(841,·)$, $\chi_{3720}(469,·)$, $\chi_{3720}(2999,·)$, $\chi_{3720}(721,·)$, $\chi_{3720}(2389,·)$, $\chi_{3720}(3611,·)$, $\chi_{3720}(349,·)$, $\chi_{3720}(481,·)$, $\chi_{3720}(3349,·)$, $\chi_{3720}(3239,·)$, $\chi_{3720}(1331,·)$, $\chi_{3720}(3371,·)$, $\chi_{3720}(109,·)$, $\chi_{3720}(3251,·)$, $\chi_{3720}(2761,·)$, $\chi_{3720}(371,·)$, $\chi_{3720}(2879,·)$, $\chi_{3720}(959,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{4} + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{5} + \frac{1}{5} a$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{4} + \frac{2}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{5} + \frac{2}{5} a$, $\frac{1}{185} a^{14} + \frac{14}{185} a^{12} - \frac{6}{185} a^{10} - \frac{3}{37} a^{8} - \frac{16}{185} a^{6} + \frac{68}{185} a^{4} + \frac{21}{185} a^{2} - \frac{67}{185}$, $\frac{1}{185} a^{15} + \frac{14}{185} a^{13} - \frac{6}{185} a^{11} - \frac{3}{37} a^{9} - \frac{16}{185} a^{7} + \frac{68}{185} a^{5} + \frac{21}{185} a^{3} - \frac{67}{185} a$, $\frac{1}{925} a^{16} - \frac{54}{925} a^{12} - \frac{1}{185} a^{10} + \frac{46}{925} a^{8} - \frac{23}{185} a^{6} + \frac{216}{925} a^{4} + \frac{24}{185} a^{2} - \frac{209}{925}$, $\frac{1}{925} a^{17} - \frac{54}{925} a^{13} - \frac{1}{185} a^{11} + \frac{46}{925} a^{9} - \frac{23}{185} a^{7} + \frac{216}{925} a^{5} + \frac{24}{185} a^{3} - \frac{209}{925} a$, $\frac{1}{23853359890573765765237860037242309564486201125} a^{18} - \frac{8427636215352453877190675157557389553205366}{23853359890573765765237860037242309564486201125} a^{16} + \frac{31531137010085741479240735291095074359673916}{23853359890573765765237860037242309564486201125} a^{14} + \frac{557450820125168395129117724636720587032332969}{23853359890573765765237860037242309564486201125} a^{12} + \frac{1151575676961263842005237969160299166100677771}{23853359890573765765237860037242309564486201125} a^{10} - \frac{2048560439030322648443530749029813700602168911}{23853359890573765765237860037242309564486201125} a^{8} + \frac{9508218173988162710780250282245692507838338581}{23853359890573765765237860037242309564486201125} a^{6} - \frac{215109713901681042702124856634308729920844358}{644685402447939615276698919925467826067194625} a^{4} - \frac{371460522431690936088986536022990930972013744}{23853359890573765765237860037242309564486201125} a^{2} - \frac{490068247672955936190841663214310954419295846}{23853359890573765765237860037242309564486201125}$, $\frac{1}{13342830068630356782335336531172193458777081837090125} a^{19} + \frac{4201200040788223779627481126824740115386012663479}{13342830068630356782335336531172193458777081837090125} a^{17} + \frac{19738622372046556462514280751661309712904084385391}{13342830068630356782335336531172193458777081837090125} a^{15} + \frac{1020488605010154688575817712377739463867601102292039}{13342830068630356782335336531172193458777081837090125} a^{13} + \frac{1170427886531152801562458060382007953804853582682046}{13342830068630356782335336531172193458777081837090125} a^{11} - \frac{345800416143920518736363873501001629819239784773641}{13342830068630356782335336531172193458777081837090125} a^{9} + \frac{4562825629113916158914615071450326404728989148635981}{13342830068630356782335336531172193458777081837090125} a^{7} + \frac{3290509064783343683611391848284304974831466028418499}{13342830068630356782335336531172193458777081837090125} a^{5} - \frac{5927087491993947905438700377253376611662114271038269}{13342830068630356782335336531172193458777081837090125} a^{3} - \frac{6609355154884644185193631451114681152119623795487026}{13342830068630356782335336531172193458777081837090125} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{22}\times C_{44}\times C_{13332}$, which has order $412972032$ (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:  \( 2725468819.573966 \) (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

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-186}) \), \(\Q(\sqrt{-465}) \), \(\Q(\sqrt{10}, \sqrt{-186})\), 5.5.923521.1, 10.10.87336042233958400000.1, 10.0.210528769967490760704.1, 10.0.20559450192137769600000.1

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 R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
$31$31.10.9.1$x^{10} - 31$$10$$1$$9$$C_{10}$$[\ ]_{10}$
31.10.9.1$x^{10} - 31$$10$$1$$9$$C_{10}$$[\ ]_{10}$