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Properties

Label 20.0.24151638874...3344.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 11^{18}\cdot 23^{10}$
Root discriminant $83.01$
Ramified primes $2, 11, 23$
Class number $155136$ (GRH)
Class group $[2, 4, 4, 4, 1212]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1039093441, -727361688, 996273093, -603969256, 479298068, -247872072, 146223201, -65291864, 31117982, -12039754, 4807824, -1609742, 544146, -155870, 44444, -10596, 2472, -462, 83, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 83*x^18 - 462*x^17 + 2472*x^16 - 10596*x^15 + 44444*x^14 - 155870*x^13 + 544146*x^12 - 1609742*x^11 + 4807824*x^10 - 12039754*x^9 + 31117982*x^8 - 65291864*x^7 + 146223201*x^6 - 247872072*x^5 + 479298068*x^4 - 603969256*x^3 + 996273093*x^2 - 727361688*x + 1039093441)
 
gp: K = bnfinit(x^20 - 10*x^19 + 83*x^18 - 462*x^17 + 2472*x^16 - 10596*x^15 + 44444*x^14 - 155870*x^13 + 544146*x^12 - 1609742*x^11 + 4807824*x^10 - 12039754*x^9 + 31117982*x^8 - 65291864*x^7 + 146223201*x^6 - 247872072*x^5 + 479298068*x^4 - 603969256*x^3 + 996273093*x^2 - 727361688*x + 1039093441, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 83 x^{18} - 462 x^{17} + 2472 x^{16} - 10596 x^{15} + 44444 x^{14} - 155870 x^{13} + 544146 x^{12} - 1609742 x^{11} + 4807824 x^{10} - 12039754 x^{9} + 31117982 x^{8} - 65291864 x^{7} + 146223201 x^{6} - 247872072 x^{5} + 479298068 x^{4} - 603969256 x^{3} + 996273093 x^{2} - 727361688 x + 1039093441 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(241516388741909997129558450172507193344=2^{20}\cdot 11^{18}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.01$
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:  $2, 11, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1012=2^{2}\cdot 11\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{1012}(1,·)$, $\chi_{1012}(139,·)$, $\chi_{1012}(967,·)$, $\chi_{1012}(137,·)$, $\chi_{1012}(459,·)$, $\chi_{1012}(783,·)$, $\chi_{1012}(597,·)$, $\chi_{1012}(919,·)$, $\chi_{1012}(93,·)$, $\chi_{1012}(415,·)$, $\chi_{1012}(229,·)$, $\chi_{1012}(873,·)$, $\chi_{1012}(875,·)$, $\chi_{1012}(45,·)$, $\chi_{1012}(1011,·)$, $\chi_{1012}(183,·)$, $\chi_{1012}(185,·)$, $\chi_{1012}(553,·)$, $\chi_{1012}(827,·)$, $\chi_{1012}(829,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{44144469852373879247009472833} a^{18} - \frac{9}{44144469852373879247009472833} a^{17} + \frac{15486781485494630746866128821}{44144469852373879247009472833} a^{16} + \frac{8539157673164591766099388135}{44144469852373879247009472833} a^{15} + \frac{7555701521887606058136504798}{44144469852373879247009472833} a^{14} - \frac{3675055597911141702152198914}{44144469852373879247009472833} a^{13} - \frac{5283666047110357408962494231}{44144469852373879247009472833} a^{12} + \frac{4492459734570179845676249316}{44144469852373879247009472833} a^{11} + \frac{13802313910762808739856764773}{44144469852373879247009472833} a^{10} - \frac{1391310649421742353360107185}{44144469852373879247009472833} a^{9} - \frac{15170082429589334414471978490}{44144469852373879247009472833} a^{8} - \frac{2921827663049896385783975151}{44144469852373879247009472833} a^{7} - \frac{5491243656736881014111328245}{44144469852373879247009472833} a^{6} - \frac{9944136898344417747458747125}{44144469852373879247009472833} a^{5} + \frac{4591810116442709602000718337}{44144469852373879247009472833} a^{4} - \frac{9086924165373554116538213868}{44144469852373879247009472833} a^{3} - \frac{15603343142774598452804291403}{44144469852373879247009472833} a^{2} + \frac{14099365807989396837007580440}{44144469852373879247009472833} a - \frac{5323247695048922421068872185}{44144469852373879247009472833}$, $\frac{1}{1335676178354856650579097575854455523} a^{19} + \frac{15128456}{1335676178354856650579097575854455523} a^{18} + \frac{263645577842558837607319066662047691}{1335676178354856650579097575854455523} a^{17} - \frac{3181904720120872104362241015995282}{1335676178354856650579097575854455523} a^{16} - \frac{172288867895442155822397381201242180}{1335676178354856650579097575854455523} a^{15} + \frac{1679377122026519250593749929066669}{19935465348579950008643247400812769} a^{14} - \frac{298624209635560421438247781200908825}{1335676178354856650579097575854455523} a^{13} - \frac{190926943378748793575503051315186160}{1335676178354856650579097575854455523} a^{12} + \frac{485511344152173621410949664224364328}{1335676178354856650579097575854455523} a^{11} - \frac{660508166364001595770081907460271399}{1335676178354856650579097575854455523} a^{10} + \frac{587413932896728726530085090742476174}{1335676178354856650579097575854455523} a^{9} + \frac{482390520021879910446615910672887205}{1335676178354856650579097575854455523} a^{8} + \frac{572996775409580843390610995306400088}{1335676178354856650579097575854455523} a^{7} + \frac{198951323978549366378769960095663386}{1335676178354856650579097575854455523} a^{6} - \frac{577527865479640691473635929465256359}{1335676178354856650579097575854455523} a^{5} - \frac{580548723178778758610374090186093917}{1335676178354856650579097575854455523} a^{4} + \frac{636422513505635199877160776131799166}{1335676178354856650579097575854455523} a^{3} + \frac{276596568644033280440629121989726534}{1335676178354856650579097575854455523} a^{2} - \frac{19447179369923963404607419562851606}{1335676178354856650579097575854455523} a + \frac{30609038272576519793875867112468582}{1335676178354856650579097575854455523}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{1212}$, which has order $155136$ (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:  $9$
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:  \( 281202.490766 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-253}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{11}, \sqrt{-23})\), \(\Q(\zeta_{11})^+\), 10.0.1379687283212183.1, 10.0.15540797558102029312.1, \(\Q(\zeta_{44})^+\)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
11Data not computed
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$