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Properties

Label 20.0.14653272768...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 11^{16}\cdot 7919^{5}$
Root discriminant $287.27$
Ramified primes $2, 5, 11, 7919$
Class number $100362425088$ (GRH)
Class group $[2, 2, 2, 4, 3136325784]$ (GRH)
Galois group 20T130

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31142373916142055599, 0, 7161290933362126941, 0, 609669460579370325, 0, 24949694443593628, 0, 558955523693880, 0, 7402326813897, 0, 60210051505, 0, 302386742, 0, 908011, 0, 1482, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 1482*x^18 + 908011*x^16 + 302386742*x^14 + 60210051505*x^12 + 7402326813897*x^10 + 558955523693880*x^8 + 24949694443593628*x^6 + 609669460579370325*x^4 + 7161290933362126941*x^2 + 31142373916142055599)
 
gp: K = bnfinit(x^20 + 1482*x^18 + 908011*x^16 + 302386742*x^14 + 60210051505*x^12 + 7402326813897*x^10 + 558955523693880*x^8 + 24949694443593628*x^6 + 609669460579370325*x^4 + 7161290933362126941*x^2 + 31142373916142055599, 1)
 

Normalized defining polynomial

\( x^{20} + 1482 x^{18} + 908011 x^{16} + 302386742 x^{14} + 60210051505 x^{12} + 7402326813897 x^{10} + 558955523693880 x^{8} + 24949694443593628 x^{6} + 609669460579370325 x^{4} + 7161290933362126941 x^{2} + 31142373916142055599 \)

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:  \(14653272768072533214308458573932333455360000000000=2^{20}\cdot 5^{10}\cdot 11^{16}\cdot 7919^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $287.27$
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:  $2, 5, 11, 7919$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7919} a^{12} + \frac{1482}{7919} a^{10} - \frac{2674}{7919} a^{8} - \frac{273}{7919} a^{6} + \frac{1864}{7919} a^{4} - \frac{2934}{7919} a^{2}$, $\frac{1}{7919} a^{13} + \frac{1482}{7919} a^{11} - \frac{2674}{7919} a^{9} - \frac{273}{7919} a^{7} + \frac{1864}{7919} a^{5} - \frac{2934}{7919} a^{3}$, $\frac{1}{62710561} a^{14} + \frac{1482}{62710561} a^{12} + \frac{908011}{62710561} a^{10} - \frac{11166063}{62710561} a^{8} + \frac{7912945}{62710561} a^{6} - \frac{27806543}{62710561} a^{4} - \frac{1420}{7919} a^{2}$, $\frac{1}{62710561} a^{15} + \frac{1482}{62710561} a^{13} + \frac{908011}{62710561} a^{11} - \frac{11166063}{62710561} a^{9} + \frac{7912945}{62710561} a^{7} - \frac{27806543}{62710561} a^{5} - \frac{1420}{7919} a^{3}$, $\frac{1}{496604932559} a^{16} + \frac{1482}{496604932559} a^{14} + \frac{908011}{496604932559} a^{12} + \frac{302386742}{496604932559} a^{10} + \frac{60210051505}{496604932559} a^{8} - \frac{46747174488}{496604932559} a^{6} - \frac{27987166}{62710561} a^{4} - \frac{3731}{7919} a^{2}$, $\frac{1}{496604932559} a^{17} + \frac{1482}{496604932559} a^{15} + \frac{908011}{496604932559} a^{13} + \frac{302386742}{496604932559} a^{11} + \frac{60210051505}{496604932559} a^{9} - \frac{46747174488}{496604932559} a^{7} - \frac{27987166}{62710561} a^{5} - \frac{3731}{7919} a^{3}$, $\frac{1}{16857841814085607403800809758296071730914697792906672347371449} a^{18} + \frac{943305247306667776866066604994017340641197072}{189413952967254015773042806272989570010277503291086206150241} a^{16} - \frac{12906117316708556841192130798123462531162998141007556}{16857841814085607403800809758296071730914697792906672347371449} a^{14} - \frac{898060785177183380778213720539207467468411124308626326411}{16857841814085607403800809758296071730914697792906672347371449} a^{12} - \frac{2483131082799552425464382270468248549322335502169374626001845}{16857841814085607403800809758296071730914697792906672347371449} a^{10} - \frac{3845344679969273364520340427385801705540315610350609421593535}{16857841814085607403800809758296071730914697792906672347371449} a^{8} - \frac{253545732401666891536888460065779490362570025315867161}{2128784166445966334613058436456127254819383481867239846871} a^{6} - \frac{95207608734740529757814792618822372951377667905817863}{268819821498417266651478524618781065136934395992832409} a^{4} + \frac{10818739021731712189268811646618768229691554721044}{33946182788030971922146549389920579004537744158711} a^{2} + \frac{863690353782274724471630507129443643819231758}{4286675437306600823607343021836163531321851769}$, $\frac{1}{16857841814085607403800809758296071730914697792906672347371449} a^{19} + \frac{943305247306667776866066604994017340641197072}{189413952967254015773042806272989570010277503291086206150241} a^{17} - \frac{12906117316708556841192130798123462531162998141007556}{16857841814085607403800809758296071730914697792906672347371449} a^{15} - \frac{898060785177183380778213720539207467468411124308626326411}{16857841814085607403800809758296071730914697792906672347371449} a^{13} - \frac{2483131082799552425464382270468248549322335502169374626001845}{16857841814085607403800809758296071730914697792906672347371449} a^{11} - \frac{3845344679969273364520340427385801705540315610350609421593535}{16857841814085607403800809758296071730914697792906672347371449} a^{9} - \frac{253545732401666891536888460065779490362570025315867161}{2128784166445966334613058436456127254819383481867239846871} a^{7} - \frac{95207608734740529757814792618822372951377667905817863}{268819821498417266651478524618781065136934395992832409} a^{5} + \frac{10818739021731712189268811646618768229691554721044}{33946182788030971922146549389920579004537744158711} a^{3} + \frac{863690353782274724471630507129443643819231758}{4286675437306600823607343021836163531321851769} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{3136325784}$, which has order $100362425088$ (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:  \( 140644.599182 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T130:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n130
Character table for t20n130 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1

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 $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
5Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
7919Data not computed