↑

Properties

Label 20.10.2634297396...6875.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,5^{15}\cdot 71^{5}\cdot 263^{4}$
Root discriminant $29.58$
Ramified primes $5, 71, 263$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1036

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -12, -176, 538, 508, -3055, -722, 3953, -562, -1052, 560, -393, 503, -317, 67, 95, -105, 36, -4, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 4*x^18 + 36*x^17 - 105*x^16 + 95*x^15 + 67*x^14 - 317*x^13 + 503*x^12 - 393*x^11 + 560*x^10 - 1052*x^9 - 562*x^8 + 3953*x^7 - 722*x^6 - 3055*x^5 + 508*x^4 + 538*x^3 - 176*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 - 4*x^18 + 36*x^17 - 105*x^16 + 95*x^15 + 67*x^14 - 317*x^13 + 503*x^12 - 393*x^11 + 560*x^10 - 1052*x^9 - 562*x^8 + 3953*x^7 - 722*x^6 - 3055*x^5 + 508*x^4 + 538*x^3 - 176*x^2 - 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 4 x^{18} + 36 x^{17} - 105 x^{16} + 95 x^{15} + 67 x^{14} - 317 x^{13} + 503 x^{12} - 393 x^{11} + 560 x^{10} - 1052 x^{9} - 562 x^{8} + 3953 x^{7} - 722 x^{6} - 3055 x^{5} + 508 x^{4} + 538 x^{3} - 176 x^{2} - 12 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-263429739612717160369873046875=-\,5^{15}\cdot 71^{5}\cdot 263^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.58$
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:  $5, 71, 263$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{13} a^{18} - \frac{3}{13} a^{17} + \frac{3}{13} a^{16} + \frac{2}{13} a^{15} - \frac{6}{13} a^{14} + \frac{5}{13} a^{13} - \frac{1}{13} a^{12} + \frac{4}{13} a^{11} + \frac{2}{13} a^{10} - \frac{1}{13} a^{9} + \frac{2}{13} a^{8} - \frac{6}{13} a^{7} - \frac{2}{13} a^{6} - \frac{2}{13} a^{5} + \frac{5}{13} a^{4} - \frac{1}{13} a^{3} - \frac{3}{13} a^{2} - \frac{2}{13} a - \frac{2}{13}$, $\frac{1}{1125321185688398759207647914412193} a^{19} - \frac{4393064209255686941688334383199}{1125321185688398759207647914412193} a^{18} - \frac{159028702481422995556566097755046}{1125321185688398759207647914412193} a^{17} + \frac{428267137233457673673801621570495}{1125321185688398759207647914412193} a^{16} - \frac{202759121389896398363864499307877}{1125321185688398759207647914412193} a^{15} - \frac{288309366079539626203409211220379}{1125321185688398759207647914412193} a^{14} - \frac{285462907065861060952184426554452}{1125321185688398759207647914412193} a^{13} + \frac{143149665776621723578237079705398}{1125321185688398759207647914412193} a^{12} + \frac{10243878736911934161815109868583}{86563168129876827631357531877861} a^{11} - \frac{230129174129982620598084163997451}{1125321185688398759207647914412193} a^{10} + \frac{297729262962514330038490096839189}{1125321185688398759207647914412193} a^{9} - \frac{282170900529783147299502600075984}{1125321185688398759207647914412193} a^{8} - \frac{366925167034906496784532684679278}{1125321185688398759207647914412193} a^{7} - \frac{288468562298503919153212023524958}{1125321185688398759207647914412193} a^{6} - \frac{335269004632111219908817391880945}{1125321185688398759207647914412193} a^{5} - \frac{491313040945562663894579682675828}{1125321185688398759207647914412193} a^{4} - \frac{973885067059409631202127610482}{5891733956483763137212816305823} a^{3} + \frac{287077579758227658326464598463682}{1125321185688398759207647914412193} a^{2} + \frac{514657927246220290835827940996554}{1125321185688398759207647914412193} a + \frac{99810055479977397323040555249708}{1125321185688398759207647914412193}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $14$
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:  \( 18658688.4911 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1036:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 396 conjugacy class representatives for t20n1036 are not computed
Character table for t20n1036 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.1089627903125.1

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 $20$ $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$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}$
71Data not computed
263Data not computed