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Properties

Label 20.4.56048985368...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{20}\cdot 5^{15}\cdot 1039^{5}\cdot 67931^{5}$
Root discriminant $612.95$
Ramified primes $2, 5, 1039, 67931$
Class number Not computed
Class group Not computed
Galois group 20T1022

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43950182127057497055453625, 0, 2520112834651844975036250, 0, 47773773343213035028450, 0, 333331964079477992575, 0, 545904341348455920, 0, -2415557541551610, 0, -9101711382939, 0, -9158944433, 0, -590506, 0, 2998, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 2998*x^18 - 590506*x^16 - 9158944433*x^14 - 9101711382939*x^12 - 2415557541551610*x^10 + 545904341348455920*x^8 + 333331964079477992575*x^6 + 47773773343213035028450*x^4 + 2520112834651844975036250*x^2 + 43950182127057497055453625)
 
gp: K = bnfinit(x^20 + 2998*x^18 - 590506*x^16 - 9158944433*x^14 - 9101711382939*x^12 - 2415557541551610*x^10 + 545904341348455920*x^8 + 333331964079477992575*x^6 + 47773773343213035028450*x^4 + 2520112834651844975036250*x^2 + 43950182127057497055453625, 1)
 

Normalized defining polynomial

\( x^{20} + 2998 x^{18} - 590506 x^{16} - 9158944433 x^{14} - 9101711382939 x^{12} - 2415557541551610 x^{10} + 545904341348455920 x^{8} + 333331964079477992575 x^{6} + 47773773343213035028450 x^{4} + 2520112834651844975036250 x^{2} + 43950182127057497055453625 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(56048985368357082097694845902442652817568000000000000000=2^{20}\cdot 5^{15}\cdot 1039^{5}\cdot 67931^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $612.95$
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:  $2, 5, 1039, 67931$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{10} - \frac{1}{5} a^{8} + \frac{2}{5} a^{6} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{11} - \frac{1}{5} a^{9} + \frac{2}{5} a^{7} + \frac{1}{5} a^{5}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{4}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{5}$, $\frac{1}{1764507725} a^{16} + \frac{2998}{1764507725} a^{14} - \frac{590506}{1764507725} a^{12} - \frac{336405808}{1764507725} a^{10} - \frac{380537389}{1764507725} a^{8} + \frac{119748328}{352901545} a^{6} - \frac{171833846}{352901545} a^{4} + \frac{7135962}{70580309} a^{2}$, $\frac{1}{1764507725} a^{17} + \frac{2998}{1764507725} a^{15} - \frac{590506}{1764507725} a^{13} - \frac{336405808}{1764507725} a^{11} - \frac{380537389}{1764507725} a^{9} + \frac{119748328}{352901545} a^{7} - \frac{171833846}{352901545} a^{5} + \frac{7135962}{70580309} a^{3}$, $\frac{1}{9535165280969517519272092739407197943518504287793846969953633113944655145398820389647617118978751180958879102075} a^{18} - \frac{278945505947719152373704901270782812972062824887948028370492920166287320793191130950239281920518285021}{9535165280969517519272092739407197943518504287793846969953633113944655145398820389647617118978751180958879102075} a^{16} - \frac{72036239884367444563555113564003995232474153344815173723576416322127706961160929590361765559634758622560938453}{9535165280969517519272092739407197943518504287793846969953633113944655145398820389647617118978751180958879102075} a^{14} - \frac{652548503920698073960381985052688534769252501081575816611084689436866705139725058074001672457283166665898863589}{9535165280969517519272092739407197943518504287793846969953633113944655145398820389647617118978751180958879102075} a^{12} - \frac{2894071211466706047820445794049530895733517775482559159497637565334935235768326169740345124907472182652741434697}{9535165280969517519272092739407197943518504287793846969953633113944655145398820389647617118978751180958879102075} a^{10} - \frac{3189819826908931672671755206731015216715485448447864226717200756761616727264358901379644272711008318113503399949}{9535165280969517519272092739407197943518504287793846969953633113944655145398820389647617118978751180958879102075} a^{8} - \frac{690461708371922978298342282652525906277237758728942800514783196004081306934737877306868279723174467817125505476}{1907033056193903503854418547881439588703700857558769393990726622788931029079764077929523423795750236191775820415} a^{6} - \frac{858226968440730997324424805901687756593391037822376540388889743356392287249028676464812727140558083069127112679}{1907033056193903503854418547881439588703700857558769393990726622788931029079764077929523423795750236191775820415} a^{4} + \frac{799027716791464500263606446727398323881308163375349593030327444945364434298083708792042732024588864307}{5403867121618590544437595329545637406330144736427179410593758162177870400311690553606172009804463270887} a^{2} - \frac{31517761067051867825223142363653611307161603908187112808556801048479896588761131424257968371315}{76563381461231496513249826230509098597600992883541773819575629261949963980912729549061226266443}$, $\frac{1}{9535165280969517519272092739407197943518504287793846969953633113944655145398820389647617118978751180958879102075} a^{19} - \frac{278945505947719152373704901270782812972062824887948028370492920166287320793191130950239281920518285021}{9535165280969517519272092739407197943518504287793846969953633113944655145398820389647617118978751180958879102075} a^{17} - \frac{72036239884367444563555113564003995232474153344815173723576416322127706961160929590361765559634758622560938453}{9535165280969517519272092739407197943518504287793846969953633113944655145398820389647617118978751180958879102075} a^{15} - \frac{652548503920698073960381985052688534769252501081575816611084689436866705139725058074001672457283166665898863589}{9535165280969517519272092739407197943518504287793846969953633113944655145398820389647617118978751180958879102075} a^{13} - \frac{2894071211466706047820445794049530895733517775482559159497637565334935235768326169740345124907472182652741434697}{9535165280969517519272092739407197943518504287793846969953633113944655145398820389647617118978751180958879102075} a^{11} - \frac{3189819826908931672671755206731015216715485448447864226717200756761616727264358901379644272711008318113503399949}{9535165280969517519272092739407197943518504287793846969953633113944655145398820389647617118978751180958879102075} a^{9} - \frac{690461708371922978298342282652525906277237758728942800514783196004081306934737877306868279723174467817125505476}{1907033056193903503854418547881439588703700857558769393990726622788931029079764077929523423795750236191775820415} a^{7} - \frac{858226968440730997324424805901687756593391037822376540388889743356392287249028676464812727140558083069127112679}{1907033056193903503854418547881439588703700857558769393990726622788931029079764077929523423795750236191775820415} a^{5} + \frac{799027716791464500263606446727398323881308163375349593030327444945364434298083708792042732024588864307}{5403867121618590544437595329545637406330144736427179410593758162177870400311690553606172009804463270887} a^{3} - \frac{31517761067051867825223142363653611307161603908187112808556801048479896588761131424257968371315}{76563381461231496513249826230509098597600992883541773819575629261949963980912729549061226266443} a$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1022:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 189 conjugacy class representatives for t20n1022 are not computed
Character table for t20n1022 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.1102817328125.1

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 32 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
1039Data not computed
67931Data not computed