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Properties

Label 20.4.42441592211...0625.1
Degree $20$
Signature $[4, 8]$
Discriminant $3^{14}\cdot 5^{14}\cdot 7^{14}\cdot 11^{8}$
Root discriminant $67.82$
Ramified primes $3, 5, 7, 11$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_5^2:Q_8$ (as 20T47)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![53361, 0, -395577, 0, 847702, 0, -80742, 0, 43057, 0, -43302, 0, -12663, 0, -2142, 0, -98, 0, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 3*x^18 - 98*x^16 - 2142*x^14 - 12663*x^12 - 43302*x^10 + 43057*x^8 - 80742*x^6 + 847702*x^4 - 395577*x^2 + 53361)
 
gp: K = bnfinit(x^20 + 3*x^18 - 98*x^16 - 2142*x^14 - 12663*x^12 - 43302*x^10 + 43057*x^8 - 80742*x^6 + 847702*x^4 - 395577*x^2 + 53361, 1)
 

Normalized defining polynomial

\( x^{20} + 3 x^{18} - 98 x^{16} - 2142 x^{14} - 12663 x^{12} - 43302 x^{10} + 43057 x^{8} - 80742 x^{6} + 847702 x^{4} - 395577 x^{2} + 53361 \)

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:  \(4244159221123694514715246343994140625=3^{14}\cdot 5^{14}\cdot 7^{14}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.82$
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:  $3, 5, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{28} a^{10} - \frac{1}{14} a^{8} - \frac{1}{14} a^{6} - \frac{1}{7} a^{4} - \frac{1}{2} a^{3} + \frac{1}{7} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{56} a^{11} - \frac{1}{56} a^{10} + \frac{3}{14} a^{9} - \frac{3}{14} a^{8} + \frac{3}{14} a^{7} - \frac{3}{14} a^{6} - \frac{1}{14} a^{5} + \frac{1}{14} a^{4} - \frac{3}{7} a^{3} - \frac{1}{14} a^{2} + \frac{1}{8} a + \frac{3}{8}$, $\frac{1}{112} a^{12} - \frac{1}{112} a^{10} - \frac{1}{28} a^{8} - \frac{5}{28} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{11}{112} a^{2} - \frac{1}{16}$, $\frac{1}{224} a^{13} - \frac{1}{224} a^{12} - \frac{1}{224} a^{11} + \frac{1}{224} a^{10} + \frac{13}{56} a^{9} - \frac{13}{56} a^{8} - \frac{5}{56} a^{7} + \frac{5}{56} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{11}{224} a^{3} - \frac{11}{224} a^{2} - \frac{1}{32} a + \frac{1}{32}$, $\frac{1}{448} a^{14} - \frac{1}{224} a^{12} + \frac{5}{448} a^{10} + \frac{3}{56} a^{8} - \frac{5}{28} a^{6} - \frac{7}{64} a^{4} - \frac{1}{2} a^{3} + \frac{1}{32} a^{2} + \frac{17}{64}$, $\frac{1}{896} a^{15} - \frac{1}{896} a^{14} - \frac{1}{448} a^{13} + \frac{1}{448} a^{12} + \frac{5}{896} a^{11} - \frac{5}{896} a^{10} - \frac{25}{112} a^{9} + \frac{25}{112} a^{8} - \frac{5}{56} a^{7} + \frac{5}{56} a^{6} - \frac{7}{128} a^{5} - \frac{57}{128} a^{4} + \frac{1}{64} a^{3} + \frac{31}{64} a^{2} + \frac{17}{128} a - \frac{17}{128}$, $\frac{1}{8960} a^{16} + \frac{1}{1792} a^{14} - \frac{9}{8960} a^{12} + \frac{31}{1792} a^{10} + \frac{127}{1120} a^{8} - \frac{429}{1792} a^{6} - \frac{3849}{8960} a^{4} - \frac{507}{1792} a^{2} - \frac{1}{2} a - \frac{137}{1280}$, $\frac{1}{17920} a^{17} - \frac{1}{17920} a^{16} + \frac{1}{3584} a^{15} - \frac{1}{3584} a^{14} - \frac{9}{17920} a^{13} + \frac{9}{17920} a^{12} + \frac{31}{3584} a^{11} - \frac{31}{3584} a^{10} + \frac{127}{2240} a^{9} - \frac{127}{2240} a^{8} + \frac{467}{3584} a^{7} - \frac{467}{3584} a^{6} - \frac{3849}{17920} a^{5} - \frac{5111}{17920} a^{4} - \frac{507}{3584} a^{3} - \frac{1285}{3584} a^{2} - \frac{137}{2560} a + \frac{137}{2560}$, $\frac{1}{1124318208722472960} a^{18} - \frac{3732187212667}{93693184060206080} a^{16} + \frac{377490523781213}{562159104361236480} a^{14} - \frac{300512958388527}{93693184060206080} a^{12} - \frac{110774582558023}{12089443104542720} a^{10} + \frac{576236268610637}{374772736240824320} a^{8} + \frac{3408113226106013}{35134944022577280} a^{6} + \frac{4374491507706431}{187386368120412160} a^{4} - \frac{1}{2} a^{3} - \frac{1230566866935863}{2509638858755520} a^{2} - \frac{1}{2} a - \frac{26582717173723669}{53538962320117760}$, $\frac{1}{24735000591894405120} a^{19} - \frac{1}{2248636417444945920} a^{18} - \frac{2837803158163}{412250009864906752} a^{17} - \frac{6724641365481}{187386368120412160} a^{16} + \frac{63785666436773}{12367500295947202560} a^{15} - \frac{691195381125653}{1124318208722472960} a^{14} - \frac{41280300237039}{412250009864906752} a^{13} + \frac{394624415591859}{187386368120412160} a^{12} - \frac{319911154120983}{265967748299939840} a^{11} - \frac{14051712714991}{3454126601297920} a^{10} - \frac{345679525631338455}{1649000039459627008} a^{9} + \frac{144313580510208051}{749545472481648640} a^{8} - \frac{46815827256923597}{220848219570485760} a^{7} + \frac{10006196522883569}{140539776090309120} a^{6} - \frac{170412136539954213}{824500019729813504} a^{5} - \frac{111264193233535287}{374772736240824320} a^{4} + \frac{758393941556699881}{1545937536993400320} a^{3} - \frac{15932970436746511}{140539776090309120} a^{2} - \frac{4170475022579713}{235571434208518144} a + \frac{32313059234548773}{107077924640235520}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 44976855810.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:Q_8$ (as 20T47):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 8 conjugacy class representatives for $C_5^2:Q_8$
Character table for $C_5^2:Q_8$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{5}, \sqrt{21})\), 10.2.98101701371953125.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 25 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ R R R R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$