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Properties

Label 40.0.11302165783...0000.7
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 $1936$ (GRH)
Class group $[22, 88]$ (GRH)
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![98029801, 0, 42588510, 0, 15501805, 0, -7579633, 0, 19241108, 0, -19221829, 0, 13197033, 0, -9636845, 0, 11066638, 0, -13370001, 0, 12917486, 0, -9643510, 0, 5579574, 0, -2503946, 0, 867354, 0, -229240, 0, 45284, 0, -6459, 0, 627, 0, -37, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 37*x^38 + 627*x^36 - 6459*x^34 + 45284*x^32 - 229240*x^30 + 867354*x^28 - 2503946*x^26 + 5579574*x^24 - 9643510*x^22 + 12917486*x^20 - 13370001*x^18 + 11066638*x^16 - 9636845*x^14 + 13197033*x^12 - 19221829*x^10 + 19241108*x^8 - 7579633*x^6 + 15501805*x^4 + 42588510*x^2 + 98029801)
 
gp: K = bnfinit(x^40 - 37*x^38 + 627*x^36 - 6459*x^34 + 45284*x^32 - 229240*x^30 + 867354*x^28 - 2503946*x^26 + 5579574*x^24 - 9643510*x^22 + 12917486*x^20 - 13370001*x^18 + 11066638*x^16 - 9636845*x^14 + 13197033*x^12 - 19221829*x^10 + 19241108*x^8 - 7579633*x^6 + 15501805*x^4 + 42588510*x^2 + 98029801, 1)
 

Normalized defining polynomial

\( x^{40} - 37 x^{38} + 627 x^{36} - 6459 x^{34} + 45284 x^{32} - 229240 x^{30} + 867354 x^{28} - 2503946 x^{26} + 5579574 x^{24} - 9643510 x^{22} + 12917486 x^{20} - 13370001 x^{18} + 11066638 x^{16} - 9636845 x^{14} + 13197033 x^{12} - 19221829 x^{10} + 19241108 x^{8} - 7579633 x^{6} + 15501805 x^{4} + 42588510 x^{2} + 98029801 \)

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$$\chi_{660}(1,·)$, $\chi_{660}(259,·)$, $\chi_{660}(389,·)$, $\chi_{660}(139,·)$, $\chi_{660}(269,·)$, $\chi_{660}(19,·)$, $\chi_{660}(409,·)$, $\chi_{660}(281,·)$, $\chi_{660}(461,·)$, $\chi_{660}(31,·)$, $\chi_{660}(161,·)$, $\chi_{660}(419,·)$, $\chi_{660}(421,·)$, $\chi_{660}(41,·)$, $\chi_{660}(301,·)$, $\chi_{660}(431,·)$, $\chi_{660}(179,·)$, $\chi_{660}(181,·)$, $\chi_{660}(439,·)$, $\chi_{660}(59,·)$, $\chi_{660}(449,·)$, $\chi_{660}(331,·)$, $\chi_{660}(589,·)$, $\chi_{660}(79,·)$, $\chi_{660}(469,·)$, $\chi_{660}(599,·)$, $\chi_{660}(89,·)$, $\chi_{660}(91,·)$, $\chi_{660}(349,·)$, $\chi_{660}(611,·)$, $\chi_{660}(131,·)$, $\chi_{660}(101,·)$, $\chi_{660}(361,·)$, $\chi_{660}(491,·)$, $\chi_{660}(109,·)$, $\chi_{660}(371,·)$, $\chi_{660}(631,·)$, $\chi_{660}(119,·)$, $\chi_{660}(509,·)$, $\chi_{660}(511,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{17711} a^{22} - \frac{22}{17711} a^{20} + \frac{209}{17711} a^{18} - \frac{1122}{17711} a^{16} + \frac{3740}{17711} a^{14} - \frac{8008}{17711} a^{12} - \frac{6700}{17711} a^{10} + \frac{8273}{17711} a^{8} + \frac{4719}{17711} a^{6} - \frac{1210}{17711} a^{4} + \frac{121}{17711} a^{2} + \frac{6763}{17711}$, $\frac{1}{17711} a^{23} - \frac{22}{17711} a^{21} + \frac{209}{17711} a^{19} - \frac{1122}{17711} a^{17} + \frac{3740}{17711} a^{15} - \frac{8008}{17711} a^{13} - \frac{6700}{17711} a^{11} + \frac{8273}{17711} a^{9} + \frac{4719}{17711} a^{7} - \frac{1210}{17711} a^{5} + \frac{121}{17711} a^{3} + \frac{6763}{17711} a$, $\frac{1}{17711} a^{24} - \frac{275}{17711} a^{20} + \frac{3476}{17711} a^{18} - \frac{3233}{17711} a^{16} + \frac{3428}{17711} a^{14} - \frac{5766}{17711} a^{12} + \frac{2561}{17711} a^{10} - \frac{8096}{17711} a^{8} - \frac{3658}{17711} a^{6} - \frac{8788}{17711} a^{4} - \frac{8286}{17711} a^{2} + \frac{7098}{17711}$, $\frac{1}{17711} a^{25} - \frac{275}{17711} a^{21} + \frac{3476}{17711} a^{19} - \frac{3233}{17711} a^{17} + \frac{3428}{17711} a^{15} - \frac{5766}{17711} a^{13} + \frac{2561}{17711} a^{11} - \frac{8096}{17711} a^{9} - \frac{3658}{17711} a^{7} - \frac{8788}{17711} a^{5} - \frac{8286}{17711} a^{3} + \frac{7098}{17711} a$, $\frac{1}{17711} a^{26} - \frac{2574}{17711} a^{20} + \frac{1109}{17711} a^{18} - \frac{4035}{17711} a^{16} - \frac{4504}{17711} a^{14} - \frac{3475}{17711} a^{12} - \frac{8652}{17711} a^{10} + \frac{4409}{17711} a^{8} - \frac{3966}{17711} a^{6} - \frac{4527}{17711} a^{4} + \frac{4951}{17711} a^{2} + \frac{170}{17711}$, $\frac{1}{17711} a^{27} - \frac{2574}{17711} a^{21} + \frac{1109}{17711} a^{19} - \frac{4035}{17711} a^{17} - \frac{4504}{17711} a^{15} - \frac{3475}{17711} a^{13} - \frac{8652}{17711} a^{11} + \frac{4409}{17711} a^{9} - \frac{3966}{17711} a^{7} - \frac{4527}{17711} a^{5} + \frac{4951}{17711} a^{3} + \frac{170}{17711} a$, $\frac{1}{17711} a^{28} - \frac{2386}{17711} a^{20} + \frac{2601}{17711} a^{18} - \frac{5639}{17711} a^{16} + \frac{6212}{17711} a^{14} - \frac{5640}{17711} a^{12} - \frac{8588}{17711} a^{10} + \frac{2114}{17711} a^{8} - \frac{7567}{17711} a^{6} + \frac{7547}{17711} a^{4} - \frac{7174}{17711} a^{2} - \frac{1951}{17711}$, $\frac{1}{17711} a^{29} - \frac{2386}{17711} a^{21} + \frac{2601}{17711} a^{19} - \frac{5639}{17711} a^{17} + \frac{6212}{17711} a^{15} - \frac{5640}{17711} a^{13} - \frac{8588}{17711} a^{11} + \frac{2114}{17711} a^{9} - \frac{7567}{17711} a^{7} + \frac{7547}{17711} a^{5} - \frac{7174}{17711} a^{3} - \frac{1951}{17711} a$, $\frac{1}{141688} a^{30} + \frac{1}{70844} a^{28} - \frac{3}{141688} a^{26} + \frac{1}{141688} a^{24} + \frac{1}{70844} a^{22} + \frac{2948}{17711} a^{20} + \frac{19159}{70844} a^{18} + \frac{8271}{70844} a^{16} - \frac{60055}{141688} a^{14} + \frac{24989}{141688} a^{12} - \frac{10823}{141688} a^{10} - \frac{51925}{141688} a^{8} - \frac{337}{1592} a^{6} + \frac{27837}{141688} a^{4} - \frac{51577}{141688} a^{2} + \frac{18009}{141688}$, $\frac{1}{1402852888} a^{31} - \frac{7579}{701426444} a^{29} + \frac{37293}{1402852888} a^{27} + \frac{18681}{1402852888} a^{25} - \frac{13883}{701426444} a^{23} - \frac{84499720}{175356611} a^{21} - \frac{329693445}{701426444} a^{19} + \frac{259714203}{701426444} a^{17} + \frac{163554329}{1402852888} a^{15} - \frac{239857355}{1402852888} a^{13} - \frac{402383471}{1402852888} a^{11} - \frac{68392909}{1402852888} a^{9} + \frac{148116815}{1402852888} a^{7} + \frac{40750869}{1402852888} a^{5} + \frac{580258359}{1402852888} a^{3} + \frac{426481593}{1402852888} a$, $\frac{1}{27777890035288} a^{32} + \frac{10885843}{27777890035288} a^{30} - \frac{97764785}{27777890035288} a^{28} + \frac{84885663}{13888945017644} a^{26} - \frac{676652205}{27777890035288} a^{24} - \frac{357394939}{13888945017644} a^{22} - \frac{1624875595217}{13888945017644} a^{20} - \frac{1726970568017}{6944472508822} a^{18} + \frac{197849593599}{27777890035288} a^{16} - \frac{5859039386601}{13888945017644} a^{14} - \frac{1102640942689}{13888945017644} a^{12} + \frac{401807157557}{6944472508822} a^{10} + \frac{5080510856661}{13888945017644} a^{8} - \frac{2260111094679}{6944472508822} a^{6} - \frac{2060039329751}{6944472508822} a^{4} - \frac{887420207224}{3472236254411} a^{2} + \frac{1277038133}{2805564088}$, $\frac{1}{27777890035288} a^{33} - \frac{4707}{27777890035288} a^{31} + \frac{299443275}{27777890035288} a^{29} + \frac{30433831}{3472236254411} a^{27} - \frac{231327715}{27777890035288} a^{25} + \frac{269940343}{13888945017644} a^{23} - \frac{56561598225}{156055561996} a^{21} + \frac{1094882395755}{3472236254411} a^{19} + \frac{6059912278419}{27777890035288} a^{17} + \frac{1318707478758}{3472236254411} a^{15} + \frac{235606658334}{3472236254411} a^{13} - \frac{5050090427009}{13888945017644} a^{11} + \frac{42422721244}{3472236254411} a^{9} + \frac{6394173897529}{13888945017644} a^{7} + \frac{5492188262207}{13888945017644} a^{5} + \frac{706204395347}{13888945017644} a^{3} + \frac{2393535124227}{27777890035288} a$, $\frac{1}{27777890035288} a^{34} - \frac{21965087}{27777890035288} a^{30} - \frac{2773801}{27777890035288} a^{28} - \frac{2528476}{3472236254411} a^{26} - \frac{104713577}{6944472508822} a^{24} - \frac{165612741}{13888945017644} a^{22} + \frac{1539101842905}{13888945017644} a^{20} - \frac{2167319500571}{27777890035288} a^{18} + \frac{3263671426579}{27777890035288} a^{16} + \frac{3085814392991}{27777890035288} a^{14} - \frac{3793737077303}{27777890035288} a^{12} - \frac{18315648879}{27777890035288} a^{10} + \frac{7495768920831}{27777890035288} a^{8} + \frac{6876657377149}{27777890035288} a^{6} + \frac{6305809345659}{27777890035288} a^{4} + \frac{3381316498795}{13888945017644} a^{2} - \frac{34934818}{350695511}$, $\frac{1}{27777890035288} a^{35} - \frac{2889}{13888945017644} a^{31} - \frac{361686727}{27777890035288} a^{29} + \frac{204731353}{27777890035288} a^{27} + \frac{451221433}{27777890035288} a^{25} + \frac{6658792}{3472236254411} a^{23} - \frac{5748985220511}{13888945017644} a^{21} + \frac{13046077537611}{27777890035288} a^{19} + \frac{4670160811809}{27777890035288} a^{17} + \frac{959000497385}{6944472508822} a^{15} + \frac{3373377749461}{13888945017644} a^{13} + \frac{2902069201303}{13888945017644} a^{11} - \frac{5918707128525}{13888945017644} a^{9} + \frac{715675342914}{3472236254411} a^{7} + \frac{927308317169}{6944472508822} a^{5} - \frac{2028718046615}{27777890035288} a^{3} - \frac{10436389119355}{27777890035288} a$, $\frac{1}{27777890035288} a^{36} - \frac{785123}{6944472508822} a^{30} + \frac{335041905}{27777890035288} a^{28} + \frac{276644479}{13888945017644} a^{26} + \frac{568116291}{27777890035288} a^{24} - \frac{15061324}{3472236254411} a^{22} - \frac{11351102519833}{27777890035288} a^{20} - \frac{10155550873393}{27777890035288} a^{18} + \frac{56342843806}{3472236254411} a^{16} + \frac{909960864699}{27777890035288} a^{14} + \frac{7439226340491}{27777890035288} a^{12} - \frac{13690969161453}{27777890035288} a^{10} - \frac{13392189391821}{27777890035288} a^{8} + \frac{796697166519}{27777890035288} a^{6} + \frac{1613576084837}{13888945017644} a^{4} + \frac{384313245964}{3472236254411} a^{2} + \frac{966104139}{2805564088}$, $\frac{1}{27777890035288} a^{37} + \frac{7867}{27777890035288} a^{31} - \frac{335855577}{27777890035288} a^{29} + \frac{335220545}{27777890035288} a^{27} - \frac{108249167}{13888945017644} a^{25} + \frac{146219731}{13888945017644} a^{23} + \frac{9788314868791}{27777890035288} a^{21} - \frac{5234600927919}{27777890035288} a^{19} - \frac{2602799415471}{13888945017644} a^{17} - \frac{5059864044583}{13888945017644} a^{15} + \frac{4101379568903}{13888945017644} a^{13} + \frac{4344473716641}{13888945017644} a^{11} + \frac{712081863892}{3472236254411} a^{9} - \frac{1517882842388}{3472236254411} a^{7} - \frac{7831315319979}{27777890035288} a^{5} - \frac{2206353705903}{27777890035288} a^{3} - \frac{2611800849685}{13888945017644} a$, $\frac{1}{27777890035288} a^{38} + \frac{91036281}{27777890035288} a^{30} + \frac{271928409}{13888945017644} a^{28} - \frac{513006465}{27777890035288} a^{26} + \frac{22225618}{3472236254411} a^{24} + \frac{99843407}{27777890035288} a^{22} + \frac{8762145059903}{27777890035288} a^{20} + \frac{928440267907}{3472236254411} a^{18} + \frac{12692745747287}{27777890035288} a^{16} + \frac{10077539432463}{27777890035288} a^{14} + \frac{11782532028663}{27777890035288} a^{12} + \frac{3448368922087}{27777890035288} a^{10} - \frac{9435131948557}{27777890035288} a^{8} - \frac{889313283893}{13888945017644} a^{6} - \frac{2219360040983}{6944472508822} a^{4} + \frac{10865348042659}{27777890035288} a^{2} + \frac{139955966}{350695511}$, $\frac{1}{27777890035288} a^{39} - \frac{8717}{27777890035288} a^{31} + \frac{207020731}{13888945017644} a^{29} - \frac{273295559}{27777890035288} a^{27} - \frac{245397811}{13888945017644} a^{25} - \frac{201686221}{27777890035288} a^{23} + \frac{1297712825015}{27777890035288} a^{21} - \frac{3238513704933}{6944472508822} a^{19} + \frac{4535574814563}{27777890035288} a^{17} + \frac{13307786695625}{27777890035288} a^{15} - \frac{412434152015}{27777890035288} a^{13} - \frac{10486000896423}{27777890035288} a^{11} + \frac{6897061350033}{27777890035288} a^{9} + \frac{1697192846107}{6944472508822} a^{7} + \frac{131283018415}{13888945017644} a^{5} + \frac{1106791392353}{27777890035288} a^{3} + \frac{2289884056065}{13888945017644} a$

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

Class group and class number

$C_{22}\times C_{88}$, which has order $1936$ (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:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{1}{7881196} a^{33} - \frac{33}{7881196} a^{31} + \frac{495}{7881196} a^{29} - \frac{2233}{3940598} a^{27} + \frac{27027}{7881196} a^{25} - \frac{57915}{3940598} a^{23} + \frac{180895}{3940598} a^{21} - \frac{208725}{1970299} a^{19} + \frac{1427679}{7881196} a^{17} - \frac{898909}{3940598} a^{15} + \frac{820743}{3940598} a^{13} - \frac{264537}{1970299} a^{11} + \frac{230945}{3940598} a^{9} - \frac{31977}{1970299} a^{7} + \frac{5049}{1970299} a^{5} - \frac{374}{1970299} a^{3} + \frac{33}{7881196} a \) (order $4$)
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:  \( 100039126568469490 \) (assuming GRH)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{55}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{15}) \), \(\Q(i, \sqrt{33})\), \(\Q(i, \sqrt{55})\), \(\Q(i, \sqrt{15})\), \(\Q(\sqrt{-15}, \sqrt{-33})\), \(\Q(\sqrt{15}, \sqrt{-33})\), \(\Q(\sqrt{15}, \sqrt{33})\), \(\Q(\sqrt{-15}, \sqrt{33})\), \(\Q(\zeta_{11})^+\), 8.0.189747360000.5, 10.0.219503494144.1, 10.0.586732839846912.1, \(\Q(\zeta_{33})^+\), 10.10.7545432611200000.1, 10.0.7368586534375.1, 10.0.162778775259375.1, 10.10.166685465865600000.1, 20.0.344255425354822086003595935744.1, 20.0.56933553290160450365440000000000.2, 20.0.27784044530832102757263360000000000.3, 20.0.3361869388230684433628866560000000000.2, 20.0.3361869388230684433628866560000000000.8, 20.20.3361869388230684433628866560000000000.3, 20.0.3206128490667995866421572265625.1

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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
$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}$
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}$
3Data not computed
5Data not computed
11Data not computed