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Properties

Label 20.0.66186603327...4481.3
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 11^{18}\cdot 17^{10}$
Root discriminant $61.81$
Ramified primes $3, 11, 17$
Class number $29110$ (GRH)
Class group $[29110]$ (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![74991181, 21109951, 2367428, 31792177, 21771895, 7639032, 21527189, -563649, 10282204, -642587, 2727803, -158015, 433108, -21161, 41961, -1600, 2419, -63, 76, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 76*x^18 - 63*x^17 + 2419*x^16 - 1600*x^15 + 41961*x^14 - 21161*x^13 + 433108*x^12 - 158015*x^11 + 2727803*x^10 - 642587*x^9 + 10282204*x^8 - 563649*x^7 + 21527189*x^6 + 7639032*x^5 + 21771895*x^4 + 31792177*x^3 + 2367428*x^2 + 21109951*x + 74991181)
 
gp: K = bnfinit(x^20 - x^19 + 76*x^18 - 63*x^17 + 2419*x^16 - 1600*x^15 + 41961*x^14 - 21161*x^13 + 433108*x^12 - 158015*x^11 + 2727803*x^10 - 642587*x^9 + 10282204*x^8 - 563649*x^7 + 21527189*x^6 + 7639032*x^5 + 21771895*x^4 + 31792177*x^3 + 2367428*x^2 + 21109951*x + 74991181, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 76 x^{18} - 63 x^{17} + 2419 x^{16} - 1600 x^{15} + 41961 x^{14} - 21161 x^{13} + 433108 x^{12} - 158015 x^{11} + 2727803 x^{10} - 642587 x^{9} + 10282204 x^{8} - 563649 x^{7} + 21527189 x^{6} + 7639032 x^{5} + 21771895 x^{4} + 31792177 x^{3} + 2367428 x^{2} + 21109951 x + 74991181 \)

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:  \(661866033279225680306281949177084481=3^{10}\cdot 11^{18}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.81$
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:  $3, 11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(561=3\cdot 11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{561}(256,·)$, $\chi_{561}(1,·)$, $\chi_{561}(392,·)$, $\chi_{561}(203,·)$, $\chi_{561}(460,·)$, $\chi_{561}(271,·)$, $\chi_{561}(152,·)$, $\chi_{561}(475,·)$, $\chi_{561}(545,·)$, $\chi_{561}(35,·)$, $\chi_{561}(356,·)$, $\chi_{561}(103,·)$, $\chi_{561}(424,·)$, $\chi_{561}(494,·)$, $\chi_{561}(239,·)$, $\chi_{561}(373,·)$, $\chi_{561}(118,·)$, $\chi_{561}(509,·)$, $\chi_{561}(254,·)$, $\chi_{561}(511,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7589} a^{11} + \frac{44}{7589} a^{9} + \frac{704}{7589} a^{7} - \frac{2661}{7589} a^{5} - \frac{1098}{7589} a^{3} + \frac{3675}{7589} a - \frac{3462}{7589}$, $\frac{1}{7589} a^{12} + \frac{44}{7589} a^{10} + \frac{704}{7589} a^{8} - \frac{2661}{7589} a^{6} - \frac{1098}{7589} a^{4} + \frac{3675}{7589} a^{2} - \frac{3462}{7589} a$, $\frac{1}{7589} a^{13} - \frac{1232}{7589} a^{9} - \frac{3281}{7589} a^{7} + \frac{2151}{7589} a^{5} - \frac{1136}{7589} a^{3} - \frac{3462}{7589} a^{2} - \frac{2331}{7589} a + \frac{548}{7589}$, $\frac{1}{7589} a^{14} - \frac{1232}{7589} a^{10} - \frac{3281}{7589} a^{8} + \frac{2151}{7589} a^{6} - \frac{1136}{7589} a^{4} - \frac{3462}{7589} a^{3} - \frac{2331}{7589} a^{2} + \frac{548}{7589} a$, $\frac{1}{37945} a^{15} - \frac{1}{37945} a^{12} - \frac{7633}{37945} a^{10} - \frac{17374}{37945} a^{9} - \frac{704}{37945} a^{8} - \frac{18434}{37945} a^{7} + \frac{2661}{37945} a^{6} + \frac{6549}{37945} a^{5} - \frac{2364}{37945} a^{4} - \frac{11814}{37945} a^{3} + \frac{4462}{37945} a^{2} - \frac{14749}{37945} a - \frac{166}{37945}$, $\frac{1}{7334743797805} a^{16} - \frac{86252846}{7334743797805} a^{15} - \frac{40248441}{1466948759561} a^{14} - \frac{333117766}{7334743797805} a^{13} + \frac{13529471}{7334743797805} a^{12} - \frac{119313453}{7334743797805} a^{11} + \frac{625290885024}{7334743797805} a^{10} + \frac{96459126062}{1466948759561} a^{9} - \frac{48861926904}{1466948759561} a^{8} - \frac{62301011089}{1466948759561} a^{7} - \frac{923572893852}{7334743797805} a^{6} + \frac{836048560412}{7334743797805} a^{5} - \frac{718036675241}{1466948759561} a^{4} + \frac{1270840337721}{7334743797805} a^{3} + \frac{3264694161969}{7334743797805} a^{2} + \frac{1854006711183}{7334743797805} a - \frac{1237101752279}{7334743797805}$, $\frac{1}{7334743797805} a^{17} - \frac{40018886}{7334743797805} a^{15} - \frac{9569256}{7334743797805} a^{14} + \frac{17016335}{1466948759561} a^{13} + \frac{323920798}{7334743797805} a^{12} + \frac{10859706}{7334743797805} a^{11} - \frac{3659512640361}{7334743797805} a^{10} + \frac{723287414302}{1466948759561} a^{9} - \frac{253340026025}{1466948759561} a^{8} - \frac{2271765069772}{7334743797805} a^{7} - \frac{204978844357}{1466948759561} a^{6} - \frac{2639107223008}{7334743797805} a^{5} + \frac{3354846819626}{7334743797805} a^{4} + \frac{507871275457}{1466948759561} a^{3} - \frac{1928270794798}{7334743797805} a^{2} + \frac{1856478223904}{7334743797805} a - \frac{893931384279}{7334743797805}$, $\frac{1}{7334743797805} a^{18} - \frac{18413443}{7334743797805} a^{15} - \frac{1602051}{1466948759561} a^{14} + \frac{139199542}{7334743797805} a^{13} - \frac{200889572}{7334743797805} a^{12} - \frac{228539689}{7334743797805} a^{11} + \frac{745963572427}{7334743797805} a^{10} + \frac{767811640544}{7334743797805} a^{9} - \frac{311930398078}{7334743797805} a^{8} + \frac{1388903608934}{7334743797805} a^{7} - \frac{3417960702806}{7334743797805} a^{6} + \frac{1654349954269}{7334743797805} a^{5} - \frac{2841515634061}{7334743797805} a^{4} + \frac{2061475870202}{7334743797805} a^{3} - \frac{1033931907334}{7334743797805} a^{2} + \frac{3068064975383}{7334743797805} a + \frac{586423390787}{7334743797805}$, $\frac{1}{7334743797805} a^{19} - \frac{51351816}{7334743797805} a^{15} + \frac{319522157}{7334743797805} a^{14} + \frac{59862495}{1466948759561} a^{13} + \frac{100146462}{7334743797805} a^{12} + \frac{141601713}{7334743797805} a^{11} + \frac{491915996723}{1466948759561} a^{10} - \frac{2030572760281}{7334743797805} a^{9} - \frac{2968942707369}{7334743797805} a^{8} + \frac{2776695394421}{7334743797805} a^{7} + \frac{126189555412}{1466948759561} a^{6} + \frac{1594845416618}{7334743797805} a^{5} + \frac{3232889547459}{7334743797805} a^{4} + \frac{3421175720396}{7334743797805} a^{3} - \frac{862953520051}{7334743797805} a^{2} + \frac{1499563938258}{7334743797805} a - \frac{76312814084}{7334743797805}$

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

Class group and class number

$C_{29110}$, which has order $29110$ (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:  \( 125582.779517 \) (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

\(\Q(\sqrt{-187}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{33}, \sqrt{-51})\), \(\Q(\zeta_{11})^+\), 10.0.3347948534700187.1, \(\Q(\zeta_{33})^+\), 10.0.73959226721104131.3

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$17$17.10.5.2$x^{10} - 83521 x^{2} + 8519142$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
17.10.5.2$x^{10} - 83521 x^{2} + 8519142$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$