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Properties

Label 20.0.99465402919...7136.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 11^{16}\cdot 17^{10}$
Root discriminant $79.41$
Ramified primes $2, 11, 17$
Class number $19910$ (GRH)
Class group $[19910]$ (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![420418912, 230512192, -162530192, -122025696, 28421816, 33068752, -1740120, -6490784, 16678, 733364, 166397, -144650, 9401, -736, 5426, -2140, 344, -84, 41, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 41*x^18 - 84*x^17 + 344*x^16 - 2140*x^15 + 5426*x^14 - 736*x^13 + 9401*x^12 - 144650*x^11 + 166397*x^10 + 733364*x^9 + 16678*x^8 - 6490784*x^7 - 1740120*x^6 + 33068752*x^5 + 28421816*x^4 - 122025696*x^3 - 162530192*x^2 + 230512192*x + 420418912)
 
gp: K = bnfinit(x^20 - 10*x^19 + 41*x^18 - 84*x^17 + 344*x^16 - 2140*x^15 + 5426*x^14 - 736*x^13 + 9401*x^12 - 144650*x^11 + 166397*x^10 + 733364*x^9 + 16678*x^8 - 6490784*x^7 - 1740120*x^6 + 33068752*x^5 + 28421816*x^4 - 122025696*x^3 - 162530192*x^2 + 230512192*x + 420418912, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 41 x^{18} - 84 x^{17} + 344 x^{16} - 2140 x^{15} + 5426 x^{14} - 736 x^{13} + 9401 x^{12} - 144650 x^{11} + 166397 x^{10} + 733364 x^{9} + 16678 x^{8} - 6490784 x^{7} - 1740120 x^{6} + 33068752 x^{5} + 28421816 x^{4} - 122025696 x^{3} - 162530192 x^{2} + 230512192 x + 420418912 \)

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:  \(99465402919592410684515977523034587136=2^{30}\cdot 11^{16}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.41$
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:  $2, 11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1496=2^{3}\cdot 11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{1496}(1,·)$, $\chi_{1496}(67,·)$, $\chi_{1496}(577,·)$, $\chi_{1496}(713,·)$, $\chi_{1496}(1291,·)$, $\chi_{1496}(273,·)$, $\chi_{1496}(339,·)$, $\chi_{1496}(203,·)$, $\chi_{1496}(137,·)$, $\chi_{1496}(1395,·)$, $\chi_{1496}(883,·)$, $\chi_{1496}(1123,·)$, $\chi_{1496}(169,·)$, $\chi_{1496}(1259,·)$, $\chi_{1496}(817,·)$, $\chi_{1496}(851,·)$, $\chi_{1496}(1257,·)$, $\chi_{1496}(441,·)$, $\chi_{1496}(1225,·)$, $\chi_{1496}(443,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{1}{8} a^{6}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} + \frac{1}{8} a^{7}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{6}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{7}$, $\frac{1}{24656} a^{16} - \frac{1}{3082} a^{15} - \frac{99}{6164} a^{14} - \frac{85}{12328} a^{13} - \frac{18}{1541} a^{12} - \frac{1295}{12328} a^{11} - \frac{1453}{12328} a^{10} - \frac{1419}{12328} a^{9} - \frac{1741}{24656} a^{8} + \frac{2127}{12328} a^{7} + \frac{887}{12328} a^{6} + \frac{355}{1541} a^{5} + \frac{63}{6164} a^{4} - \frac{1095}{3082} a^{3} + \frac{1347}{3082} a^{2} - \frac{190}{1541} a - \frac{493}{1541}$, $\frac{1}{24656} a^{17} - \frac{5}{268} a^{15} - \frac{16}{1541} a^{14} + \frac{717}{12328} a^{13} + \frac{635}{12328} a^{12} - \frac{513}{6164} a^{11} - \frac{715}{12328} a^{10} - \frac{2871}{24656} a^{9} + \frac{1327}{12328} a^{8} + \frac{119}{1541} a^{7} + \frac{97}{536} a^{6} - \frac{905}{6164} a^{5} - \frac{145}{6164} a^{4} + \frac{146}{1541} a^{3} - \frac{17}{134} a^{2} - \frac{472}{1541} a + \frac{679}{1541}$, $\frac{1}{21442104974640575716573648} a^{18} - \frac{9}{21442104974640575716573648} a^{17} + \frac{63252480652628343203}{10721052487320287858286824} a^{16} - \frac{253009922610513372761}{5360526243660143929143412} a^{15} + \frac{8643352217944615735181}{1340131560915035982285853} a^{14} - \frac{237586188456765256561929}{5360526243660143929143412} a^{13} + \frac{18319065815908442724746}{1340131560915035982285853} a^{12} - \frac{85624405905409226240991}{10721052487320287858286824} a^{11} + \frac{1725419357568368741283649}{21442104974640575716573648} a^{10} + \frac{789717464730864621917477}{21442104974640575716573648} a^{9} - \frac{53289851165060549663855}{10721052487320287858286824} a^{8} + \frac{126204243052051498974497}{10721052487320287858286824} a^{7} - \frac{2250790808950406341755757}{10721052487320287858286824} a^{6} + \frac{322315686972131845111588}{1340131560915035982285853} a^{5} - \frac{293008252890569003254649}{1340131560915035982285853} a^{4} - \frac{1821688523958915694503}{62331700507676092199342} a^{3} + \frac{335877719485850949570982}{1340131560915035982285853} a^{2} - \frac{167787823823986806379568}{1340131560915035982285853} a - \frac{8591419656869351157390}{31165850253838046099671}$, $\frac{1}{482309982362840431454290992637264} a^{19} + \frac{11246787}{482309982362840431454290992637264} a^{18} + \frac{2712528154916155106183230575}{482309982362840431454290992637264} a^{17} + \frac{7708635912527963489425674155}{482309982362840431454290992637264} a^{16} - \frac{2366587919814846174852337682269}{241154991181420215727145496318632} a^{15} - \frac{380908932682877031716924572365}{30144373897677526965893187039829} a^{14} - \frac{6021570911714813734931485150871}{120577495590710107863572748159316} a^{13} + \frac{814012788405483987440254268758}{30144373897677526965893187039829} a^{12} - \frac{30063241793169301597292651083505}{482309982362840431454290992637264} a^{11} - \frac{19415675597733651392215174642663}{482309982362840431454290992637264} a^{10} + \frac{44504130135041322326689534483661}{482309982362840431454290992637264} a^{9} + \frac{467300163373975066068935620571}{11216511217740475150099790526448} a^{8} + \frac{44273552728058802883871445973057}{241154991181420215727145496318632} a^{7} - \frac{20229917451423713851357286162501}{120577495590710107863572748159316} a^{6} - \frac{193136065471890222205469737005}{60288747795355053931786374079658} a^{5} + \frac{29608398820932392210872305898149}{120577495590710107863572748159316} a^{4} + \frac{14099766596115243004609894744939}{30144373897677526965893187039829} a^{3} - \frac{8733267566155181202345081667792}{30144373897677526965893187039829} a^{2} - \frac{2842660892151245389365358826238}{30144373897677526965893187039829} a - \frac{2218524031629263576147245518}{7876763495604266257092549527}$

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

Class group and class number

$C_{19910}$, which has order $19910$ (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:  \( 3338983.62101 \) (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

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{-2}, \sqrt{17})\), \(\Q(\zeta_{11})^+\), 10.0.7024111812608.1, 10.10.304358957700017.1, 10.0.9973234325914157056.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.10.15.9$x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.9$x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$$2$$5$$15$$C_{10}$$[3]^{5}$
$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}$
$17$17.10.5.1$x^{10} - 578 x^{6} + 83521 x^{2} - 51114852$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
17.10.5.1$x^{10} - 578 x^{6} + 83521 x^{2} - 51114852$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$