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Properties

Label 20.0.78483048970...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{16}$
Root discriminant $139.55$
Ramified primes $2, 3, 5, 7, 11$
Class number $1060840$ (GRH)
Class group $[2, 530420]$ (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![15777305149, -2632174544, 3440960115, -1045516946, 827188384, -43081774, 58532261, -33816384, 8810954, 3714770, 64610, -866142, 112792, 80734, -8888, -6260, 890, 238, -33, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 33*x^18 + 238*x^17 + 890*x^16 - 6260*x^15 - 8888*x^14 + 80734*x^13 + 112792*x^12 - 866142*x^11 + 64610*x^10 + 3714770*x^9 + 8810954*x^8 - 33816384*x^7 + 58532261*x^6 - 43081774*x^5 + 827188384*x^4 - 1045516946*x^3 + 3440960115*x^2 - 2632174544*x + 15777305149)
 
gp: K = bnfinit(x^20 - 6*x^19 - 33*x^18 + 238*x^17 + 890*x^16 - 6260*x^15 - 8888*x^14 + 80734*x^13 + 112792*x^12 - 866142*x^11 + 64610*x^10 + 3714770*x^9 + 8810954*x^8 - 33816384*x^7 + 58532261*x^6 - 43081774*x^5 + 827188384*x^4 - 1045516946*x^3 + 3440960115*x^2 - 2632174544*x + 15777305149, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 33 x^{18} + 238 x^{17} + 890 x^{16} - 6260 x^{15} - 8888 x^{14} + 80734 x^{13} + 112792 x^{12} - 866142 x^{11} + 64610 x^{10} + 3714770 x^{9} + 8810954 x^{8} - 33816384 x^{7} + 58532261 x^{6} - 43081774 x^{5} + 827188384 x^{4} - 1045516946 x^{3} + 3440960115 x^{2} - 2632174544 x + 15777305149 \)

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:  \(7848304897073886403551554174576640000000000=2^{20}\cdot 3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $139.55$
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:  $2, 3, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4620=2^{2}\cdot 3\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{4620}(1,·)$, $\chi_{4620}(3331,·)$, $\chi_{4620}(4229,·)$, $\chi_{4620}(1651,·)$, $\chi_{4620}(449,·)$, $\chi_{4620}(841,·)$, $\chi_{4620}(1681,·)$, $\chi_{4620}(3389,·)$, $\chi_{4620}(2071,·)$, $\chi_{4620}(3359,·)$, $\chi_{4620}(3809,·)$, $\chi_{4620}(419,·)$, $\chi_{4620}(421,·)$, $\chi_{4620}(839,·)$, $\chi_{4620}(1709,·)$, $\chi_{4620}(4591,·)$, $\chi_{4620}(2099,·)$, $\chi_{4620}(1259,·)$, $\chi_{4620}(2491,·)$, $\chi_{4620}(2941,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{43} a^{15} + \frac{17}{43} a^{14} - \frac{4}{43} a^{13} - \frac{2}{43} a^{12} + \frac{6}{43} a^{11} - \frac{2}{43} a^{10} - \frac{21}{43} a^{9} + \frac{1}{43} a^{7} - \frac{2}{43} a^{6} - \frac{20}{43} a^{5} - \frac{11}{43} a^{4} + \frac{18}{43} a^{3} + \frac{14}{43} a^{2} - \frac{11}{43} a + \frac{16}{43}$, $\frac{1}{43} a^{16} + \frac{8}{43} a^{14} - \frac{20}{43} a^{13} - \frac{3}{43} a^{12} - \frac{18}{43} a^{11} + \frac{13}{43} a^{10} + \frac{13}{43} a^{9} + \frac{1}{43} a^{8} - \frac{19}{43} a^{7} + \frac{14}{43} a^{6} - \frac{15}{43} a^{5} - \frac{10}{43} a^{4} + \frac{9}{43} a^{3} + \frac{9}{43} a^{2} - \frac{12}{43} a - \frac{14}{43}$, $\frac{1}{43} a^{17} + \frac{16}{43} a^{14} - \frac{14}{43} a^{13} - \frac{2}{43} a^{12} + \frac{8}{43} a^{11} - \frac{14}{43} a^{10} - \frac{3}{43} a^{9} - \frac{19}{43} a^{8} + \frac{6}{43} a^{7} + \frac{1}{43} a^{6} + \frac{21}{43} a^{5} + \frac{11}{43} a^{4} - \frac{6}{43} a^{3} + \frac{5}{43} a^{2} - \frac{12}{43} a + \frac{1}{43}$, $\frac{1}{2421962595992219291677062090671} a^{18} + \frac{24116042787542094517248632763}{2421962595992219291677062090671} a^{17} + \frac{1855430310564257175934538645}{2421962595992219291677062090671} a^{16} - \frac{6112300404246607698727301558}{2421962595992219291677062090671} a^{15} - \frac{457494560329134923321962443090}{2421962595992219291677062090671} a^{14} - \frac{1041107629255835187721304388183}{2421962595992219291677062090671} a^{13} + \frac{1160716634013666117875511346979}{2421962595992219291677062090671} a^{12} + \frac{1201918052977337975099653156900}{2421962595992219291677062090671} a^{11} + \frac{293457567704902463582912523670}{2421962595992219291677062090671} a^{10} + \frac{770633408571657874736429801825}{2421962595992219291677062090671} a^{9} + \frac{1108933476120848539192306186661}{2421962595992219291677062090671} a^{8} + \frac{612412384038177187219133774709}{2421962595992219291677062090671} a^{7} - \frac{802522503696837013830994516360}{2421962595992219291677062090671} a^{6} - \frac{787044348676203827803638508311}{2421962595992219291677062090671} a^{5} - \frac{138487679015171557805137211345}{2421962595992219291677062090671} a^{4} + \frac{622655595544425515348732328904}{2421962595992219291677062090671} a^{3} - \frac{724988759742510478549321155236}{2421962595992219291677062090671} a^{2} - \frac{1035480753653595863833430601131}{2421962595992219291677062090671} a - \frac{918682494503517258240259231920}{2421962595992219291677062090671}$, $\frac{1}{18344492492731593915266897949542694134927178049231622979719} a^{19} + \frac{3676610375844780761748181016}{18344492492731593915266897949542694134927178049231622979719} a^{18} - \frac{131687285391619325454826379452679684166743881830424486955}{18344492492731593915266897949542694134927178049231622979719} a^{17} + \frac{79464934297729158389356637653106193982081857260675504545}{18344492492731593915266897949542694134927178049231622979719} a^{16} - \frac{195219511224443032169861740071448334316984099636947899974}{18344492492731593915266897949542694134927178049231622979719} a^{15} + \frac{1140485200378870780551620217925575951624916375245029523827}{18344492492731593915266897949542694134927178049231622979719} a^{14} + \frac{171000959462504494963821087687884264246549977032077713927}{426616104482130091052718556966109165928539024400735418133} a^{13} - \frac{7397887362638574697172583045646596372062898513924094809406}{18344492492731593915266897949542694134927178049231622979719} a^{12} + \frac{1743277465769373197599161204652656481872802329044618028332}{18344492492731593915266897949542694134927178049231622979719} a^{11} - \frac{5916706044257025628629109109768098053886458586245400768369}{18344492492731593915266897949542694134927178049231622979719} a^{10} + \frac{273400784470149509444113623053767972266052714472782150267}{797586630118764952837691215197508440649007741270940129553} a^{9} + \frac{4613238402137179713700780827561249894655110264475502170762}{18344492492731593915266897949542694134927178049231622979719} a^{8} - \frac{6480562730745308290430000124257184265537300698468236436379}{18344492492731593915266897949542694134927178049231622979719} a^{7} - \frac{58109834384343745384616261677619837264919635314640671267}{797586630118764952837691215197508440649007741270940129553} a^{6} + \frac{3708706113924971031666807094722974824391168307803084780365}{18344492492731593915266897949542694134927178049231622979719} a^{5} + \frac{231458025705666801582535559015181789596062602881554546168}{18344492492731593915266897949542694134927178049231622979719} a^{4} + \frac{3111024926140988990291682301817590409965751970102436598097}{18344492492731593915266897949542694134927178049231622979719} a^{3} + \frac{1331443538609731445931563620176363249560905868567543822145}{18344492492731593915266897949542694134927178049231622979719} a^{2} + \frac{1283945583587171994962302668635242941510590904497922122486}{18344492492731593915266897949542694134927178049231622979719} a - \frac{6953576944764321032216312087982340692090956527750541199471}{18344492492731593915266897949542694134927178049231622979719}$

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

Class group and class number

$C_{2}\times C_{530420}$, which has order $1060840$ (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:  \( 8013735.512306594 \) (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

\(\Q(\sqrt{-105}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{7}, \sqrt{-15})\), \(\Q(\zeta_{11})^+\), 10.0.2801482624803139200000.1, 10.0.162778775259375.1, 10.10.3689195226078208.1

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 R R R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$3$3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} - 18 x^{6} + 81 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5Data not computed
7Data not computed
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$