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Properties

Label 20.4.18530919141...0000.2
Degree $20$
Signature $[4, 8]$
Discriminant $2^{20}\cdot 5^{14}\cdot 6029^{7}$
Root discriminant $129.84$
Ramified primes $2, 5, 6029$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T375

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5478679059725, 0, 3980198089500, 0, 1051048984525, 0, 121116031230, 0, 4607526225, 0, -220957380, 0, -24058174, 0, -668642, 0, -3522, 0, 103, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 103*x^18 - 3522*x^16 - 668642*x^14 - 24058174*x^12 - 220957380*x^10 + 4607526225*x^8 + 121116031230*x^6 + 1051048984525*x^4 + 3980198089500*x^2 + 5478679059725)
 
gp: K = bnfinit(x^20 + 103*x^18 - 3522*x^16 - 668642*x^14 - 24058174*x^12 - 220957380*x^10 + 4607526225*x^8 + 121116031230*x^6 + 1051048984525*x^4 + 3980198089500*x^2 + 5478679059725, 1)
 

Normalized defining polynomial

\( x^{20} + 103 x^{18} - 3522 x^{16} - 668642 x^{14} - 24058174 x^{12} - 220957380 x^{10} + 4607526225 x^{8} + 121116031230 x^{6} + 1051048984525 x^{4} + 3980198089500 x^{2} + 5478679059725 \)

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:  \(1853091914135141046229493177600000000000000=2^{20}\cdot 5^{14}\cdot 6029^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.84$
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:  $2, 5, 6029$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{12} - \frac{2}{5} a^{10} - \frac{2}{5} a^{8} + \frac{1}{5} a^{6}$, $\frac{1}{5} a^{15} - \frac{2}{5} a^{13} - \frac{2}{5} a^{11} - \frac{2}{5} a^{9} + \frac{1}{5} a^{7}$, $\frac{1}{30145} a^{16} + \frac{103}{30145} a^{14} - \frac{3522}{30145} a^{12} - \frac{5452}{30145} a^{10} - \frac{2464}{30145} a^{8} + \frac{1094}{6029} a^{6} + \frac{2740}{6029} a^{4} - \frac{1432}{6029} a^{2}$, $\frac{1}{30145} a^{17} + \frac{103}{30145} a^{15} - \frac{3522}{30145} a^{13} - \frac{5452}{30145} a^{11} - \frac{2464}{30145} a^{9} + \frac{1094}{6029} a^{7} + \frac{2740}{6029} a^{5} - \frac{1432}{6029} a^{3}$, $\frac{1}{111296448454637398153046766223054898517770437142130715} a^{18} + \frac{240817897996952584711892716627296743079310558218}{111296448454637398153046766223054898517770437142130715} a^{16} - \frac{372296476582416075665395344335387327441521389348035}{22259289690927479630609353244610979703554087428426143} a^{14} + \frac{32310221864119356262781031241998370712934885326291564}{111296448454637398153046766223054898517770437142130715} a^{12} + \frac{16487305498425726121429903687134168430018115309439832}{111296448454637398153046766223054898517770437142130715} a^{10} + \frac{28882753960022765676601632858107076014350238939052571}{111296448454637398153046766223054898517770437142130715} a^{8} - \frac{51970572297901450641322116768003938520327547805290768}{111296448454637398153046766223054898517770437142130715} a^{6} + \frac{3199803103779269815616012411540020498417496842042810}{22259289690927479630609353244610979703554087428426143} a^{4} - \frac{737283240766696596123225851139156776326660448009}{3692036770762560894113344376283128164464104731867} a^{2} - \frac{142437654238683391269088238933696021144255409}{612379626930263873629680606449349504804130823}$, $\frac{1}{556482242273186990765233831115274492588852185710653575} a^{19} + \frac{240817897996952584711892716627296743079310558218}{556482242273186990765233831115274492588852185710653575} a^{17} + \frac{20397807308015399252282376522934043066346480481685968}{556482242273186990765233831115274492588852185710653575} a^{15} - \frac{234801254427010399304531207693333385729714163814822152}{556482242273186990765233831115274492588852185710653575} a^{13} + \frac{83265174571208165013257963420967107540680377594718261}{556482242273186990765233831115274492588852185710653575} a^{11} + \frac{3826424921312208182737187703677600605000500048973240}{22259289690927479630609353244610979703554087428426143} a^{9} - \frac{1188451304278958840428510540935718352670938415074585}{22259289690927479630609353244610979703554087428426143} a^{7} + \frac{25459092794706749446225365656151000201971584270468953}{111296448454637398153046766223054898517770437142130715} a^{5} + \frac{1329358060151685038420692580285419910520309803145}{3692036770762560894113344376283128164464104731867} a^{3} - \frac{273439381619842227705689890366479006150503411}{612379626930263873629680606449349504804130823} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  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:  \( 6986915812620 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T375:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 48 conjugacy class representatives for t20n375
Character table for t20n375 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1

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

Sibling fields

Degree 20 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 $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
6029Data not computed