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Properties

Label 20.0.17268259226...7744.6
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $91.59$
Ramified primes $2, 7, 11$
Class number $209312$ (GRH)
Class group $[2, 4, 26164]$ (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![558235129, 0, 275545100, 0, 125864335, 0, 47451488, 0, 13405291, 0, 2689136, 0, 371596, 0, 34328, 0, 2017, 0, 68, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 68*x^18 + 2017*x^16 + 34328*x^14 + 371596*x^12 + 2689136*x^10 + 13405291*x^8 + 47451488*x^6 + 125864335*x^4 + 275545100*x^2 + 558235129)
 
gp: K = bnfinit(x^20 + 68*x^18 + 2017*x^16 + 34328*x^14 + 371596*x^12 + 2689136*x^10 + 13405291*x^8 + 47451488*x^6 + 125864335*x^4 + 275545100*x^2 + 558235129, 1)
 

Normalized defining polynomial

\( x^{20} + 68 x^{18} + 2017 x^{16} + 34328 x^{14} + 371596 x^{12} + 2689136 x^{10} + 13405291 x^{8} + 47451488 x^{6} + 125864335 x^{4} + 275545100 x^{2} + 558235129 \)

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:  \(1726825922665179584778586849482316447744=2^{40}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.59$
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:  $2, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(616=2^{3}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{616}(1,·)$, $\chi_{616}(195,·)$, $\chi_{616}(391,·)$, $\chi_{616}(139,·)$, $\chi_{616}(141,·)$, $\chi_{616}(83,·)$, $\chi_{616}(533,·)$, $\chi_{616}(475,·)$, $\chi_{616}(477,·)$, $\chi_{616}(225,·)$, $\chi_{616}(421,·)$, $\chi_{616}(167,·)$, $\chi_{616}(169,·)$, $\chi_{616}(615,·)$, $\chi_{616}(113,·)$, $\chi_{616}(449,·)$, $\chi_{616}(307,·)$, $\chi_{616}(309,·)$, $\chi_{616}(503,·)$, $\chi_{616}(447,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{23627} a^{11} + \frac{33}{23627} a^{9} + \frac{396}{23627} a^{7} + \frac{2079}{23627} a^{5} + \frac{4455}{23627} a^{3} + \frac{2673}{23627} a$, $\frac{1}{211059991} a^{12} - \frac{101005392}{211059991} a^{10} - \frac{75322480}{211059991} a^{8} + \frac{53351845}{211059991} a^{6} - \frac{12565109}{211059991} a^{4} - \frac{18851673}{211059991} a^{2} + \frac{3739}{8933}$, $\frac{1}{211059991} a^{13} + \frac{39}{211059991} a^{11} + \frac{91956878}{211059991} a^{9} - \frac{49895769}{211059991} a^{7} - \frac{26965105}{211059991} a^{5} - \frac{19557380}{211059991} a^{3} - \frac{80930064}{211059991} a$, $\frac{1}{211059991} a^{14} + \frac{21027337}{211059991} a^{10} - \frac{67158923}{211059991} a^{8} + \frac{2912850}{211059991} a^{6} + \frac{48361889}{211059991} a^{4} + \frac{21105210}{211059991} a^{2} - \frac{2893}{8933}$, $\frac{1}{211059991} a^{15} - \frac{945}{211059991} a^{11} + \frac{83147735}{211059991} a^{9} - \frac{92947173}{211059991} a^{7} + \frac{19981748}{211059991} a^{5} + \frac{50744904}{211059991} a^{3} + \frac{76066900}{211059991} a$, $\frac{1}{211059991} a^{16} + \frac{32168227}{211059991} a^{10} + \frac{65586185}{211059991} a^{8} - \frac{5862576}{211059991} a^{6} - \frac{3923605}{211059991} a^{4} - \frac{9724841}{211059991} a^{2} - \frac{4113}{8933}$, $\frac{1}{211059991} a^{17} + \frac{494}{211059991} a^{11} + \frac{59350951}{211059991} a^{9} - \frac{80685384}{211059991} a^{7} + \frac{25376635}{211059991} a^{5} - \frac{7241467}{211059991} a^{3} + \frac{30948168}{211059991} a$, $\frac{1}{211059991} a^{18} - \frac{65203268}{211059991} a^{10} - \frac{17938680}{211059991} a^{8} + \frac{52064080}{211059991} a^{6} + \frac{79182640}{211059991} a^{4} + \frac{57035026}{211059991} a^{2} + \frac{2065}{8933}$, $\frac{1}{211059991} a^{19} - \frac{1301}{211059991} a^{11} + \frac{23126321}{211059991} a^{9} - \frac{88335881}{211059991} a^{7} - \frac{77502180}{211059991} a^{5} - \frac{97809596}{211059991} a^{3} - \frac{1905020}{211059991} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{26164}$, which has order $209312$ (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:  \( 530208.250733 \) (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

\(\Q(\sqrt{-77}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-154}) \), \(\Q(\sqrt{2}, \sqrt{-77})\), \(\Q(\zeta_{11})^+\), 10.0.40581147486860288.1, 10.10.7024111812608.1, 10.0.1298596719579529216.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$7$7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed