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Properties

Label 20.0.97243636996...256.11
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $79.32$
Ramified primes $2, 3, 7, 11$
Class number $44440$ (GRH)
Class group $[2, 22220]$ (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![1335925537, -1529298058, 2346260051, -1420801804, 778793317, -278561850, 126200543, -44565012, 24070297, -4262356, 2417591, -505438, 266341, -27244, 14571, -1898, 973, -48, 25, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 25*x^18 - 48*x^17 + 973*x^16 - 1898*x^15 + 14571*x^14 - 27244*x^13 + 266341*x^12 - 505438*x^11 + 2417591*x^10 - 4262356*x^9 + 24070297*x^8 - 44565012*x^7 + 126200543*x^6 - 278561850*x^5 + 778793317*x^4 - 1420801804*x^3 + 2346260051*x^2 - 1529298058*x + 1335925537)
 
gp: K = bnfinit(x^20 - 2*x^19 + 25*x^18 - 48*x^17 + 973*x^16 - 1898*x^15 + 14571*x^14 - 27244*x^13 + 266341*x^12 - 505438*x^11 + 2417591*x^10 - 4262356*x^9 + 24070297*x^8 - 44565012*x^7 + 126200543*x^6 - 278561850*x^5 + 778793317*x^4 - 1420801804*x^3 + 2346260051*x^2 - 1529298058*x + 1335925537, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 25 x^{18} - 48 x^{17} + 973 x^{16} - 1898 x^{15} + 14571 x^{14} - 27244 x^{13} + 266341 x^{12} - 505438 x^{11} + 2417591 x^{10} - 4262356 x^{9} + 24070297 x^{8} - 44565012 x^{7} + 126200543 x^{6} - 278561850 x^{5} + 778793317 x^{4} - 1420801804 x^{3} + 2346260051 x^{2} - 1529298058 x + 1335925537 \)

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:  \(97243636996704282094565176844674400256=2^{20}\cdot 3^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.32$
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:  $2, 3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(924=2^{2}\cdot 3\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{924}(1,·)$, $\chi_{924}(491,·)$, $\chi_{924}(769,·)$, $\chi_{924}(841,·)$, $\chi_{924}(587,·)$, $\chi_{924}(13,·)$, $\chi_{924}(335,·)$, $\chi_{924}(659,·)$, $\chi_{924}(853,·)$, $\chi_{924}(601,·)$, $\chi_{924}(349,·)$, $\chi_{924}(419,·)$, $\chi_{924}(421,·)$, $\chi_{924}(743,·)$, $\chi_{924}(169,·)$, $\chi_{924}(839,·)$, $\chi_{924}(827,·)$, $\chi_{924}(239,·)$, $\chi_{924}(757,·)$, $\chi_{924}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10} - \frac{1}{11} a^{9} + \frac{1}{11} a^{8} - \frac{1}{11} a^{7} + \frac{1}{11} a^{6} - \frac{1}{11} a^{5} + \frac{1}{11} a^{4} - \frac{1}{11} a^{3} + \frac{1}{11} a^{2} - \frac{1}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{11} + \frac{1}{11}$, $\frac{1}{11} a^{12} + \frac{1}{11} a$, $\frac{1}{11} a^{13} + \frac{1}{11} a^{2}$, $\frac{1}{11} a^{14} + \frac{1}{11} a^{3}$, $\frac{1}{4807} a^{15} - \frac{182}{4807} a^{14} - \frac{18}{437} a^{13} - \frac{1}{209} a^{12} - \frac{179}{4807} a^{11} + \frac{160}{4807} a^{10} - \frac{1062}{4807} a^{9} - \frac{1116}{4807} a^{8} + \frac{1149}{4807} a^{7} + \frac{1128}{4807} a^{6} + \frac{1248}{4807} a^{5} + \frac{821}{4807} a^{4} + \frac{1891}{4807} a^{3} - \frac{1754}{4807} a^{2} + \frac{1566}{4807} a + \frac{58}{209}$, $\frac{1}{13631589653} a^{16} - \frac{1186976}{13631589653} a^{15} - \frac{33200318}{13631589653} a^{14} - \frac{6947625}{13631589653} a^{13} - \frac{531189917}{13631589653} a^{12} - \frac{402512736}{13631589653} a^{11} + \frac{400924099}{13631589653} a^{10} + \frac{1339321221}{13631589653} a^{9} + \frac{1831974951}{13631589653} a^{8} + \frac{5760172711}{13631589653} a^{7} - \frac{1102613134}{13631589653} a^{6} - \frac{3291485420}{13631589653} a^{5} + \frac{44298446}{13631589653} a^{4} + \frac{3225579180}{13631589653} a^{3} + \frac{5187423462}{13631589653} a^{2} + \frac{5186171894}{13631589653} a + \frac{43014350}{592677811}$, $\frac{1}{13631589653} a^{17} + \frac{20740}{1239235423} a^{15} + \frac{416297494}{13631589653} a^{14} + \frac{23063236}{717452087} a^{13} - \frac{472909313}{13631589653} a^{12} + \frac{311893673}{13631589653} a^{11} - \frac{607797105}{13631589653} a^{10} + \frac{1837870070}{13631589653} a^{9} - \frac{4122598219}{13631589653} a^{8} - \frac{3912650123}{13631589653} a^{7} - \frac{1478156132}{13631589653} a^{6} - \frac{4746576601}{13631589653} a^{5} - \frac{319777351}{13631589653} a^{4} - \frac{9732642}{1239235423} a^{3} - \frac{2858163371}{13631589653} a^{2} - \frac{270619706}{13631589653} a + \frac{492425}{53879801}$, $\frac{1}{13631589653} a^{18} - \frac{901426}{13631589653} a^{15} + \frac{23356692}{1239235423} a^{14} + \frac{140472191}{13631589653} a^{13} - \frac{43911822}{1239235423} a^{12} + \frac{304755427}{13631589653} a^{11} + \frac{584262585}{13631589653} a^{10} + \frac{45446473}{13631589653} a^{9} - \frac{207000943}{592677811} a^{8} + \frac{3374270434}{13631589653} a^{7} - \frac{4979804858}{13631589653} a^{6} + \frac{4357286555}{13631589653} a^{5} + \frac{2534033212}{13631589653} a^{4} - \frac{1711342737}{13631589653} a^{3} - \frac{3933789501}{13631589653} a^{2} - \frac{5970712863}{13631589653} a - \frac{256397798}{592677811}$, $\frac{1}{1795330098088901899402272081112837541431803520225339} a^{19} - \frac{27230530846112541215618553494570817237802}{1795330098088901899402272081112837541431803520225339} a^{18} + \frac{23080430112557169187337976599968543519803}{1795330098088901899402272081112837541431803520225339} a^{17} - \frac{14166715510208374440893781731261923996158}{1795330098088901899402272081112837541431803520225339} a^{16} + \frac{127544416881989433537754524920312619374316136613}{1795330098088901899402272081112837541431803520225339} a^{15} + \frac{79240663424778823895247974522720669710060617205549}{1795330098088901899402272081112837541431803520225339} a^{14} + \frac{64247001473896195468248813296293143074343900629380}{1795330098088901899402272081112837541431803520225339} a^{13} - \frac{25862263816784876623486326238229545740701037123204}{1795330098088901899402272081112837541431803520225339} a^{12} + \frac{22235758587152136245020835184671191652821598022893}{1795330098088901899402272081112837541431803520225339} a^{11} - \frac{4805871777780974384258330262965180464450696339867}{163211827098991081763842916464803412857436683656849} a^{10} - \frac{7997739235700828313527094122493535713190560277129}{163211827098991081763842916464803412857436683656849} a^{9} - \frac{36370634635458532492463443366035727290609400093857}{94491057794152731547488004269096712706937027380281} a^{8} - \frac{734954498977014891556318556286389295155953049141184}{1795330098088901899402272081112837541431803520225339} a^{7} - \frac{22283996964630821290570245674250550378574674934977}{1795330098088901899402272081112837541431803520225339} a^{6} + \frac{786417228709052906748581287751994691638718601316580}{1795330098088901899402272081112837541431803520225339} a^{5} - \frac{440015721792273957371741067782567439513829918507847}{1795330098088901899402272081112837541431803520225339} a^{4} + \frac{108081996016829655335648685144590896437808411258722}{1795330098088901899402272081112837541431803520225339} a^{3} + \frac{518153489260243296426819145589814890226626937613696}{1795330098088901899402272081112837541431803520225339} a^{2} + \frac{868032667957427315889080816049654874491307197843898}{1795330098088901899402272081112837541431803520225339} a + \frac{28887090186819449995946218864587349689195919466328}{78057830351691386930533568744036414844861022618493}$

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

Class group and class number

$C_{2}\times C_{22220}$, which has order $44440$ (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:  \( 1415140.16249 \) (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

\(\Q(\sqrt{77}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{-21}, \sqrt{-33})\), \(\Q(\zeta_{11})^+\), 10.10.39630026842637.1, 10.0.896474439937004544.3, 10.0.586732839846912.1

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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
3Data not computed
$7$7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$