Properties

Label 20.0.285...224.2
Degree $20$
Signature $[0, 10]$
Discriminant $2.855\times 10^{35}$
Root discriminant $59.26$
Ramified primes $2, 439$
Class number $30$ (GRH)
Class group $[30]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 6*x^18 + 134*x^17 - 319*x^16 - 856*x^15 + 4445*x^14 - 3980*x^13 - 11054*x^12 + 18302*x^11 + 93567*x^10 - 220966*x^9 + 264342*x^8 - 82996*x^7 + 520317*x^6 - 864204*x^5 + 1920879*x^4 - 1481450*x^3 + 4270922*x^2 - 2667276*x + 5331177)
 
gp: K = bnfinit(x^20 - 8*x^19 + 6*x^18 + 134*x^17 - 319*x^16 - 856*x^15 + 4445*x^14 - 3980*x^13 - 11054*x^12 + 18302*x^11 + 93567*x^10 - 220966*x^9 + 264342*x^8 - 82996*x^7 + 520317*x^6 - 864204*x^5 + 1920879*x^4 - 1481450*x^3 + 4270922*x^2 - 2667276*x + 5331177, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5331177, -2667276, 4270922, -1481450, 1920879, -864204, 520317, -82996, 264342, -220966, 93567, 18302, -11054, -3980, 4445, -856, -319, 134, 6, -8, 1]);
 

\(x^{20} - 8 x^{19} + 6 x^{18} + 134 x^{17} - 319 x^{16} - 856 x^{15} + 4445 x^{14} - 3980 x^{13} - 11054 x^{12} + 18302 x^{11} + 93567 x^{10} - 220966 x^{9} + 264342 x^{8} - 82996 x^{7} + 520317 x^{6} - 864204 x^{5} + 1920879 x^{4} - 1481450 x^{3} + 4270922 x^{2} - 2667276 x + 5331177\)  Toggle raw display

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(285459875695026432742900224149684224\)\(\medspace = 2^{30}\cdot 439^{10}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $59.26$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 439$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $20$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{39} a^{12} + \frac{5}{39} a^{11} - \frac{1}{39} a^{10} - \frac{2}{39} a^{9} + \frac{1}{39} a^{8} - \frac{16}{39} a^{7} + \frac{16}{39} a^{6} - \frac{7}{39} a^{5} + \frac{2}{13} a^{4} + \frac{6}{13} a^{3} + \frac{11}{39} a^{2} - \frac{11}{39} a - \frac{3}{13}$, $\frac{1}{117} a^{13} + \frac{1}{117} a^{12} + \frac{5}{117} a^{11} - \frac{11}{117} a^{10} + \frac{1}{13} a^{9} - \frac{7}{117} a^{8} - \frac{11}{117} a^{7} + \frac{20}{117} a^{6} - \frac{19}{39} a^{5} + \frac{46}{117} a^{4} + \frac{56}{117} a^{3} + \frac{49}{117} a^{2} - \frac{56}{117} a + \frac{4}{13}$, $\frac{1}{117} a^{14} + \frac{1}{117} a^{12} + \frac{8}{117} a^{11} - \frac{16}{117} a^{10} - \frac{10}{117} a^{9} - \frac{7}{117} a^{8} - \frac{38}{117} a^{7} - \frac{8}{117} a^{6} + \frac{7}{117} a^{5} - \frac{8}{117} a^{4} + \frac{17}{117} a^{3} + \frac{2}{13} a^{2} + \frac{8}{117} a - \frac{1}{13}$, $\frac{1}{117} a^{15} + \frac{1}{117} a^{12} - \frac{4}{39} a^{11} + \frac{7}{117} a^{10} - \frac{4}{117} a^{9} + \frac{2}{117} a^{8} - \frac{19}{39} a^{7} + \frac{47}{117} a^{6} + \frac{4}{9} a^{5} - \frac{2}{9} a^{4} + \frac{10}{117} a^{3} + \frac{49}{117} a^{2} - \frac{43}{117} a + \frac{2}{13}$, $\frac{1}{351} a^{16} + \frac{1}{351} a^{15} + \frac{1}{117} a^{12} + \frac{2}{27} a^{11} - \frac{1}{351} a^{10} - \frac{2}{351} a^{9} + \frac{2}{117} a^{8} - \frac{44}{351} a^{7} + \frac{124}{351} a^{6} - \frac{61}{351} a^{5} + \frac{67}{351} a^{4} + \frac{1}{9} a^{3} + \frac{161}{351} a^{2} + \frac{139}{351} a + \frac{3}{13}$, $\frac{1}{1053} a^{17} - \frac{1}{1053} a^{15} + \frac{1}{351} a^{13} - \frac{1}{81} a^{12} + \frac{16}{117} a^{11} + \frac{152}{1053} a^{10} - \frac{154}{1053} a^{9} - \frac{86}{1053} a^{8} - \frac{103}{351} a^{7} - \frac{410}{1053} a^{6} + \frac{380}{1053} a^{5} + \frac{107}{1053} a^{4} - \frac{175}{1053} a^{3} - \frac{184}{1053} a^{2} - \frac{10}{81} a - \frac{17}{39}$, $\frac{1}{12363600483} a^{18} - \frac{4427974}{12363600483} a^{17} + \frac{13748393}{12363600483} a^{16} - \frac{23468594}{12363600483} a^{15} - \frac{4152179}{4121200161} a^{14} - \frac{44206189}{12363600483} a^{13} - \frac{99499154}{12363600483} a^{12} - \frac{1420229794}{12363600483} a^{11} - \frac{408647731}{4121200161} a^{10} + \frac{367013870}{12363600483} a^{9} + \frac{1733835575}{12363600483} a^{8} + \frac{4109624479}{12363600483} a^{7} + \frac{4580128558}{12363600483} a^{6} - \frac{1554427763}{4121200161} a^{5} - \frac{361613539}{1373733387} a^{4} - \frac{86537138}{457911129} a^{3} - \frac{11808083}{30081753} a^{2} + \frac{830668327}{12363600483} a + \frac{220218776}{457911129}$, $\frac{1}{27835553995279205541576097587245392757267777589} a^{19} + \frac{654643671371296874875277854459434413}{27835553995279205541576097587245392757267777589} a^{18} - \frac{11457088944834475872103290445913939547880393}{27835553995279205541576097587245392757267777589} a^{17} - \frac{25143399735704205214140409514799921071180642}{27835553995279205541576097587245392757267777589} a^{16} + \frac{17272775416752933916827195088526022874428118}{9278517998426401847192032529081797585755925863} a^{15} - \frac{73313443258554640192830842839381710974091904}{27835553995279205541576097587245392757267777589} a^{14} - \frac{5766260774590358288734768425374589652974056}{2141196461175323503198161352865030212097521353} a^{13} - \frac{19960965633153930760377113463073785162779413}{2141196461175323503198161352865030212097521353} a^{12} - \frac{359627111766584160635195136592683966209763797}{3092839332808800615730677509693932528585308621} a^{11} - \frac{2193821278935910132404178031255342739149826811}{27835553995279205541576097587245392757267777589} a^{10} - \frac{2022776974705518395463669957022635538656868888}{27835553995279205541576097587245392757267777589} a^{9} + \frac{1223566534342983429484905246022078914326068159}{27835553995279205541576097587245392757267777589} a^{8} + \frac{9157494064666360635932467570600440076133269084}{27835553995279205541576097587245392757267777589} a^{7} - \frac{17391046960918102924540804691087399837610643}{9278517998426401847192032529081797585755925863} a^{6} - \frac{2492093605064583477281982220799598619578293203}{9278517998426401847192032529081797585755925863} a^{5} + \frac{11021724971168792657796208428395506781461798}{237910717908369278133129039207225579121946817} a^{4} - \frac{14831604345421754029718953581670590635190475}{79303572636123092711043013069075193040648939} a^{3} - \frac{6422408873187003660642550050306909606877690142}{27835553995279205541576097587245392757267777589} a^{2} + \frac{274728977404256794529976380790318699147214765}{3092839332808800615730677509693932528585308621} a - \frac{49108212633168797204536083388029490527127870}{114549604918844467249284352210886389947604023}$  Toggle raw display

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

Class group and class number

$C_{30}$, which has order $30$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 243668061.264 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{10}\cdot 243668061.264 \cdot 30}{2\sqrt{285459875695026432742900224149684224}}\approx 0.656018390476$ (assuming GRH)

Galois group

$D_{10}$ (as 20T4):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-439}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-878}) \), \(\Q(\sqrt{2}, \sqrt{-439})\), 5.1.192721.1 x5, 10.0.16305067506199.1, 10.2.1217048865701888.3 x5, 10.0.534284452043128832.2 x5

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

Sibling fields

Degree 10 siblings: 10.2.1217048865701888.3, 10.0.534284452043128832.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.2.0.1}{2} }^{10}$ ${\href{/padicField/5.10.0.1}{10} }^{2}$ ${\href{/padicField/7.5.0.1}{5} }^{4}$ ${\href{/padicField/11.10.0.1}{10} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{10}$ ${\href{/padicField/17.2.0.1}{2} }^{10}$ ${\href{/padicField/19.10.0.1}{10} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{10}$ ${\href{/padicField/29.10.0.1}{10} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{10}$ ${\href{/padicField/37.2.0.1}{2} }^{10}$ ${\href{/padicField/41.2.0.1}{2} }^{10}$ ${\href{/padicField/43.2.0.1}{2} }^{10}$ ${\href{/padicField/47.2.0.1}{2} }^{10}$ ${\href{/padicField/53.10.0.1}{10} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{10}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
439Data not computed