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Properties

Label 40.0.11302165783...0000.4
Degree $40$
Signature $[0, 20]$
Discriminant $2^{40}\cdot 3^{20}\cdot 5^{20}\cdot 11^{36}$
Root discriminant $67.04$
Ramified primes $2, 3, 5, 11$
Class number Not computed
Class group Not computed
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14641, 0, 219615, 0, 2210791, 0, 11068596, 0, 37846985, 0, 90820785, 0, 164857660, 0, 229629444, 0, 252235148, 0, 218421093, 0, 150504519, 0, 81549039, 0, 34765478, 0, 11395659, 0, 2888875, 0, 550209, 0, 79937, 0, 8481, 0, 660, 0, 33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 33*x^38 + 660*x^36 + 8481*x^34 + 79937*x^32 + 550209*x^30 + 2888875*x^28 + 11395659*x^26 + 34765478*x^24 + 81549039*x^22 + 150504519*x^20 + 218421093*x^18 + 252235148*x^16 + 229629444*x^14 + 164857660*x^12 + 90820785*x^10 + 37846985*x^8 + 11068596*x^6 + 2210791*x^4 + 219615*x^2 + 14641)
 
gp: K = bnfinit(x^40 + 33*x^38 + 660*x^36 + 8481*x^34 + 79937*x^32 + 550209*x^30 + 2888875*x^28 + 11395659*x^26 + 34765478*x^24 + 81549039*x^22 + 150504519*x^20 + 218421093*x^18 + 252235148*x^16 + 229629444*x^14 + 164857660*x^12 + 90820785*x^10 + 37846985*x^8 + 11068596*x^6 + 2210791*x^4 + 219615*x^2 + 14641, 1)
 

Normalized defining polynomial

\( x^{40} + 33 x^{38} + 660 x^{36} + 8481 x^{34} + 79937 x^{32} + 550209 x^{30} + 2888875 x^{28} + 11395659 x^{26} + 34765478 x^{24} + 81549039 x^{22} + 150504519 x^{20} + 218421093 x^{18} + 252235148 x^{16} + 229629444 x^{14} + 164857660 x^{12} + 90820785 x^{10} + 37846985 x^{8} + 11068596 x^{6} + 2210791 x^{4} + 219615 x^{2} + 14641 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11302165783522556415463223790320401501047994110286233600000000000000000000=2^{40}\cdot 3^{20}\cdot 5^{20}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.04$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(660=2^{2}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{660}(1,·)$, $\chi_{660}(259,·)$, $\chi_{660}(389,·)$, $\chi_{660}(391,·)$, $\chi_{660}(521,·)$, $\chi_{660}(139,·)$, $\chi_{660}(269,·)$, $\chi_{660}(271,·)$, $\chi_{660}(401,·)$, $\chi_{660}(131,·)$, $\chi_{660}(151,·)$, $\chi_{660}(491,·)$, $\chi_{660}(289,·)$, $\chi_{660}(421,·)$, $\chi_{660}(169,·)$, $\chi_{660}(299,·)$, $\chi_{660}(301,·)$, $\chi_{660}(431,·)$, $\chi_{660}(49,·)$, $\chi_{660}(181,·)$, $\chi_{660}(439,·)$, $\chi_{660}(641,·)$, $\chi_{660}(571,·)$, $\chi_{660}(19,·)$, $\chi_{660}(449,·)$, $\chi_{660}(581,·)$, $\chi_{660}(79,·)$, $\chi_{660}(211,·)$, $\chi_{660}(659,·)$, $\chi_{660}(89,·)$, $\chi_{660}(221,·)$, $\chi_{660}(479,·)$, $\chi_{660}(611,·)$, $\chi_{660}(229,·)$, $\chi_{660}(529,·)$, $\chi_{660}(361,·)$, $\chi_{660}(359,·)$, $\chi_{660}(239,·)$, $\chi_{660}(371,·)$, $\chi_{660}(509,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10}$, $\frac{1}{11} a^{11}$, $\frac{1}{11} a^{12}$, $\frac{1}{11} a^{13}$, $\frac{1}{11} a^{14}$, $\frac{1}{11} a^{15}$, $\frac{1}{11} a^{16}$, $\frac{1}{11} a^{17}$, $\frac{1}{11} a^{18}$, $\frac{1}{11} a^{19}$, $\frac{1}{121} a^{20}$, $\frac{1}{121} a^{21}$, $\frac{1}{121} a^{22}$, $\frac{1}{121} a^{23}$, $\frac{1}{121} a^{24}$, $\frac{1}{121} a^{25}$, $\frac{1}{121} a^{26}$, $\frac{1}{121} a^{27}$, $\frac{1}{1331} a^{28} + \frac{1}{121} a^{18} + \frac{1}{11} a^{8}$, $\frac{1}{1331} a^{29} + \frac{1}{121} a^{19} + \frac{1}{11} a^{9}$, $\frac{1}{10648} a^{30} - \frac{1}{484} a^{24} - \frac{3}{88} a^{18} - \frac{1}{22} a^{12} + \frac{3}{8}$, $\frac{1}{10648} a^{31} - \frac{1}{484} a^{25} - \frac{3}{88} a^{19} - \frac{1}{22} a^{13} + \frac{3}{8} a$, $\frac{1}{457864} a^{32} - \frac{21}{457864} a^{30} + \frac{6}{57233} a^{28} - \frac{21}{20812} a^{26} + \frac{37}{20812} a^{24} + \frac{20}{5203} a^{22} - \frac{9}{41624} a^{20} - \frac{1459}{41624} a^{18} + \frac{17}{473} a^{16} + \frac{25}{946} a^{14} - \frac{41}{946} a^{12} - \frac{10}{473} a^{10} - \frac{225}{473} a^{8} + \frac{5}{43} a^{6} + \frac{6}{43} a^{4} + \frac{3}{344} a^{2} - \frac{7}{344}$, $\frac{1}{457864} a^{33} - \frac{21}{457864} a^{31} + \frac{6}{57233} a^{29} - \frac{21}{20812} a^{27} + \frac{37}{20812} a^{25} + \frac{20}{5203} a^{23} - \frac{9}{41624} a^{21} - \frac{1459}{41624} a^{19} + \frac{17}{473} a^{17} + \frac{25}{946} a^{15} - \frac{41}{946} a^{13} - \frac{10}{473} a^{11} - \frac{225}{473} a^{9} + \frac{5}{43} a^{7} + \frac{6}{43} a^{5} + \frac{3}{344} a^{3} - \frac{7}{344} a$, $\frac{1}{457864} a^{34} - \frac{3}{228932} a^{30} - \frac{71}{228932} a^{28} - \frac{15}{5203} a^{26} - \frac{23}{10406} a^{24} - \frac{89}{41624} a^{22} + \frac{9}{5203} a^{20} - \frac{489}{20812} a^{18} - \frac{35}{946} a^{16} - \frac{16}{473} a^{14} + \frac{1}{43} a^{12} - \frac{5}{473} a^{10} - \frac{26}{473} a^{8} - \frac{18}{43} a^{6} - \frac{21}{344} a^{4} + \frac{7}{43} a^{2} - \frac{9}{172}$, $\frac{1}{457864} a^{35} - \frac{3}{228932} a^{31} - \frac{71}{228932} a^{29} - \frac{15}{5203} a^{27} - \frac{23}{10406} a^{25} - \frac{89}{41624} a^{23} + \frac{9}{5203} a^{21} - \frac{489}{20812} a^{19} - \frac{35}{946} a^{17} - \frac{16}{473} a^{15} + \frac{1}{43} a^{13} - \frac{5}{473} a^{11} - \frac{26}{473} a^{9} - \frac{18}{43} a^{7} - \frac{21}{344} a^{5} + \frac{7}{43} a^{3} - \frac{9}{172} a$, $\frac{1}{38985140253928} a^{36} + \frac{39008515}{38985140253928} a^{34} - \frac{42160227}{38985140253928} a^{32} + \frac{1713449873}{38985140253928} a^{30} - \frac{640196767}{1772051829724} a^{28} - \frac{4947923989}{1772051829724} a^{26} - \frac{4550943855}{3544103659448} a^{24} - \frac{11962618643}{3544103659448} a^{22} - \frac{3940171149}{3544103659448} a^{20} - \frac{416356067}{29290112888} a^{18} - \frac{196573309}{80547810442} a^{16} + \frac{1489860369}{80547810442} a^{14} - \frac{867112581}{80547810442} a^{12} - \frac{619678714}{40273905221} a^{10} - \frac{32801445}{85145677} a^{8} - \frac{12361691309}{29290112888} a^{6} + \frac{12662312809}{29290112888} a^{4} + \frac{2968748199}{29290112888} a^{2} - \frac{7586464803}{29290112888}$, $\frac{1}{38985140253928} a^{37} + \frac{39008515}{38985140253928} a^{35} - \frac{42160227}{38985140253928} a^{33} + \frac{1713449873}{38985140253928} a^{31} - \frac{640196767}{1772051829724} a^{29} - \frac{4947923989}{1772051829724} a^{27} - \frac{4550943855}{3544103659448} a^{25} - \frac{11962618643}{3544103659448} a^{23} - \frac{3940171149}{3544103659448} a^{21} - \frac{416356067}{29290112888} a^{19} - \frac{196573309}{80547810442} a^{17} + \frac{1489860369}{80547810442} a^{15} - \frac{867112581}{80547810442} a^{13} - \frac{619678714}{40273905221} a^{11} - \frac{32801445}{85145677} a^{9} - \frac{12361691309}{29290112888} a^{7} + \frac{12662312809}{29290112888} a^{5} + \frac{2968748199}{29290112888} a^{3} - \frac{7586464803}{29290112888} a$, $\frac{1}{50225137859076169279549444873675592} a^{38} + \frac{26007227832222171699}{4565921623552379025413585897606872} a^{36} + \frac{1037789179998922264369501225}{1141480405888094756353396474401718} a^{34} - \frac{2849092955350646949763981955}{4565921623552379025413585897606872} a^{32} + \frac{34512606376921455363129976967}{4565921623552379025413585897606872} a^{30} - \frac{6050626175205130947742611036}{570740202944047378176698237200859} a^{28} + \frac{1421540294076452207301006682751}{415083783959307184128507808873352} a^{26} - \frac{1291182069884252050792599805789}{415083783959307184128507808873352} a^{24} - \frac{44975788726086333522013543311}{103770945989826796032126952218338} a^{22} + \frac{501810393479249864389787046597}{415083783959307184128507808873352} a^{20} + \frac{9261987578301294265923894324043}{415083783959307184128507808873352} a^{18} - \frac{180565787795794426357870798286}{4716861181355763456005770555379} a^{16} + \frac{187666586700107323045401906917}{9433722362711526912011541110758} a^{14} + \frac{203664021150317402114986193893}{9433722362711526912011541110758} a^{12} - \frac{7084855679182093531800512263}{4716861181355763456005770555379} a^{10} - \frac{60066045459650597117751307959}{563207305236509069373823349896} a^{8} + \frac{97944842560173549198492084465}{3430444495531464331640560403912} a^{6} + \frac{300515791868255678410437475179}{857611123882866082910140100978} a^{4} + \frac{1014496620188146734047797663749}{3430444495531464331640560403912} a^{2} - \frac{489354525452158429023745192549}{3430444495531464331640560403912}$, $\frac{1}{50225137859076169279549444873675592} a^{39} + \frac{26007227832222171699}{4565921623552379025413585897606872} a^{37} + \frac{1037789179998922264369501225}{1141480405888094756353396474401718} a^{35} - \frac{2849092955350646949763981955}{4565921623552379025413585897606872} a^{33} + \frac{34512606376921455363129976967}{4565921623552379025413585897606872} a^{31} - \frac{6050626175205130947742611036}{570740202944047378176698237200859} a^{29} + \frac{1421540294076452207301006682751}{415083783959307184128507808873352} a^{27} - \frac{1291182069884252050792599805789}{415083783959307184128507808873352} a^{25} - \frac{44975788726086333522013543311}{103770945989826796032126952218338} a^{23} + \frac{501810393479249864389787046597}{415083783959307184128507808873352} a^{21} + \frac{9261987578301294265923894324043}{415083783959307184128507808873352} a^{19} - \frac{180565787795794426357870798286}{4716861181355763456005770555379} a^{17} + \frac{187666586700107323045401906917}{9433722362711526912011541110758} a^{15} + \frac{203664021150317402114986193893}{9433722362711526912011541110758} a^{13} - \frac{7084855679182093531800512263}{4716861181355763456005770555379} a^{11} - \frac{60066045459650597117751307959}{563207305236509069373823349896} a^{9} + \frac{97944842560173549198492084465}{3430444495531464331640560403912} a^{7} + \frac{300515791868255678410437475179}{857611123882866082910140100978} a^{5} + \frac{1014496620188146734047797663749}{3430444495531464331640560403912} a^{3} - \frac{489354525452158429023745192549}{3430444495531464331640560403912} a$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{1234475919464273636438511}{97905515557774659606604037602} a^{38} + \frac{40511375081111345145982927}{97905515557774659606604037602} a^{36} + \frac{1614732687845248081788408567}{195811031115549319213208075204} a^{34} + \frac{20645704017829059376322914827}{195811031115549319213208075204} a^{32} + \frac{9005871720195486649401584643}{9107489819327875312242236056} a^{30} + \frac{60162866500248869263378586711}{8900501414343150873327639782} a^{28} + \frac{156722629934632381134053329620}{4450250707171575436663819891} a^{26} + \frac{2446479118466885322469253009457}{17801002828686301746655279564} a^{24} + \frac{7373325763227460306745043752349}{17801002828686301746655279564} a^{22} + \frac{17023426379753166795770511613201}{17801002828686301746655279564} a^{20} + \frac{5612767115000177019052269620733}{3236545968852054863028232648} a^{18} + \frac{996270769303849762017030052912}{404568246106506857878529081} a^{16} + \frac{1122819535478582251122504571659}{404568246106506857878529081} a^{14} + \frac{1982268403992325877385991413463}{809136492213013715757058162} a^{12} + \frac{686964718049809459686322918749}{404568246106506857878529081} a^{10} + \frac{65554027262848022033025374523}{73557862928455792341550742} a^{8} + \frac{25875592177012809967392614409}{73557862928455792341550742} a^{6} + \frac{13770790092582395198814832745}{147115725856911584683101484} a^{4} + \frac{2702152383350278895911764489}{147115725856911584683101484} a^{2} + \frac{525524317081005827943329531}{294231451713823169366202968} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_{10}$ (as 40T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{-165}) \), \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{5}, \sqrt{-33})\), \(\Q(\sqrt{-3}, \sqrt{55})\), \(\Q(\sqrt{11}, \sqrt{-15})\), \(\Q(\sqrt{-15}, \sqrt{-33})\), \(\Q(\sqrt{-3}, \sqrt{11})\), \(\Q(\sqrt{5}, \sqrt{11})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{11})^+\), 8.0.189747360000.9, 10.0.1833540124521600000.1, 10.0.586732839846912.1, 10.10.669871503125.1, 10.10.7545432611200000.1, 10.0.52089208083.1, 10.0.162778775259375.1, \(\Q(\zeta_{44})^+\), 20.0.3361869388230684433628866560000000000.6, 20.0.3361869388230684433628866560000000000.9, 20.0.3361869388230684433628866560000000000.5, 20.0.3361869388230684433628866560000000000.2, 20.0.344255425354822086003595935744.2, 20.20.56933553290160450365440000000000.1, 20.0.26496929674942114598525390625.1

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
2Data not computed
3Data not computed
5Data not computed
11Data not computed