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Properties

Label 20.0.12569066605...8944.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 11^{18}\cdot 1451^{2}$
Root discriminant $50.70$
Ramified primes $2, 11, 1451$
Class number $1700$ (GRH)
Class group $[2, 850]$ (GRH)
Galois group 20T340

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![539177, -150400, 1076236, -209964, 844391, 19962, 353348, 113984, 119267, 56452, 50458, 17820, 12527, 5494, 1790, 938, 202, 6, 32, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 32*x^18 + 6*x^17 + 202*x^16 + 938*x^15 + 1790*x^14 + 5494*x^13 + 12527*x^12 + 17820*x^11 + 50458*x^10 + 56452*x^9 + 119267*x^8 + 113984*x^7 + 353348*x^6 + 19962*x^5 + 844391*x^4 - 209964*x^3 + 1076236*x^2 - 150400*x + 539177)
 
gp: K = bnfinit(x^20 - 4*x^19 + 32*x^18 + 6*x^17 + 202*x^16 + 938*x^15 + 1790*x^14 + 5494*x^13 + 12527*x^12 + 17820*x^11 + 50458*x^10 + 56452*x^9 + 119267*x^8 + 113984*x^7 + 353348*x^6 + 19962*x^5 + 844391*x^4 - 209964*x^3 + 1076236*x^2 - 150400*x + 539177, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 32 x^{18} + 6 x^{17} + 202 x^{16} + 938 x^{15} + 1790 x^{14} + 5494 x^{13} + 12527 x^{12} + 17820 x^{11} + 50458 x^{10} + 56452 x^{9} + 119267 x^{8} + 113984 x^{7} + 353348 x^{6} + 19962 x^{5} + 844391 x^{4} - 209964 x^{3} + 1076236 x^{2} - 150400 x + 539177 \)

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:  \(12569066605710630176407970532818944=2^{30}\cdot 11^{18}\cdot 1451^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.70$
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:  $2, 11, 1451$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{36830489166195777438162606245903182135714268171932465848246163} a^{19} + \frac{3199877219855898215723806085415848301407782295341574686641704}{36830489166195777438162606245903182135714268171932465848246163} a^{18} - \frac{8285922736765851791925107005227051923848830273527487857895921}{36830489166195777438162606245903182135714268171932465848246163} a^{17} + \frac{15661446560375530734912576948226712802634576912609474007861123}{36830489166195777438162606245903182135714268171932465848246163} a^{16} + \frac{5935924415187844639223989453601632448033516724951600791640447}{36830489166195777438162606245903182135714268171932465848246163} a^{15} - \frac{6825812926453077299365290026064284836627843815674795645026210}{36830489166195777438162606245903182135714268171932465848246163} a^{14} - \frac{8200226193356263604493837984018658375699484859325237832334262}{36830489166195777438162606245903182135714268171932465848246163} a^{13} + \frac{17037514825460145178225957775811428027895953690517537289770336}{36830489166195777438162606245903182135714268171932465848246163} a^{12} + \frac{16167114962864869339173760147996986846601699018537136858615556}{36830489166195777438162606245903182135714268171932465848246163} a^{11} - \frac{4477324552798714247299528428600814568105355407631513747023421}{36830489166195777438162606245903182135714268171932465848246163} a^{10} + \frac{17854854403288307731205116483281343118570177796891099890083350}{36830489166195777438162606245903182135714268171932465848246163} a^{9} + \frac{10140317243386841744164692362675002993865403170332508957177885}{36830489166195777438162606245903182135714268171932465848246163} a^{8} + \frac{18276556592300543223281233122129334258517874760743028699849207}{36830489166195777438162606245903182135714268171932465848246163} a^{7} + \frac{18262244182298802227968391804902470673521461346142214934589108}{36830489166195777438162606245903182135714268171932465848246163} a^{6} + \frac{10823214518058247371522683345193006465555382869507907084470593}{36830489166195777438162606245903182135714268171932465848246163} a^{5} + \frac{17991160413464997949839439033580940100892724036359625180354149}{36830489166195777438162606245903182135714268171932465848246163} a^{4} - \frac{15806727410508650169183241163679299544897747935115557915659091}{36830489166195777438162606245903182135714268171932465848246163} a^{3} - \frac{5767218862661068335837269180972476369708487180344057040236730}{36830489166195777438162606245903182135714268171932465848246163} a^{2} - \frac{12029421879344767399204415236474932990403965776626992000247374}{36830489166195777438162606245903182135714268171932465848246163} a - \frac{7820119696035868484395643228744238992791921327316718430889013}{36830489166195777438162606245903182135714268171932465848246163}$

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

Class group and class number

$C_{2}\times C_{850}$, which has order $1700$ (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:  \( 281202.490766 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T340:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 80 conjugacy class representatives for t20n340 are not computed
Character table for t20n340 is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{44})^+\)

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
1451Data not computed