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Properties

Label 40.0.11302165783...000.10
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![1099511627776, 0, 481036337152, 0, 141733920768, 0, 31943819264, 0, 5117050880, 0, 242221056, 0, -213843968, 0, -108695552, 0, -34189056, 0, -8164240, 0, -1435039, 0, -510265, 0, -133551, 0, -26537, 0, -3263, 0, 231, 0, 305, 0, 119, 0, 33, 0, 7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 7*x^38 + 33*x^36 + 119*x^34 + 305*x^32 + 231*x^30 - 3263*x^28 - 26537*x^26 - 133551*x^24 - 510265*x^22 - 1435039*x^20 - 8164240*x^18 - 34189056*x^16 - 108695552*x^14 - 213843968*x^12 + 242221056*x^10 + 5117050880*x^8 + 31943819264*x^6 + 141733920768*x^4 + 481036337152*x^2 + 1099511627776)
 
gp: K = bnfinit(x^40 + 7*x^38 + 33*x^36 + 119*x^34 + 305*x^32 + 231*x^30 - 3263*x^28 - 26537*x^26 - 133551*x^24 - 510265*x^22 - 1435039*x^20 - 8164240*x^18 - 34189056*x^16 - 108695552*x^14 - 213843968*x^12 + 242221056*x^10 + 5117050880*x^8 + 31943819264*x^6 + 141733920768*x^4 + 481036337152*x^2 + 1099511627776, 1)
 

Normalized defining polynomial

\( x^{40} + 7 x^{38} + 33 x^{36} + 119 x^{34} + 305 x^{32} + 231 x^{30} - 3263 x^{28} - 26537 x^{26} - 133551 x^{24} - 510265 x^{22} - 1435039 x^{20} - 8164240 x^{18} - 34189056 x^{16} - 108695552 x^{14} - 213843968 x^{12} + 242221056 x^{10} + 5117050880 x^{8} + 31943819264 x^{6} + 141733920768 x^{4} + 481036337152 x^{2} + 1099511627776 \)

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$$\chi_{660}(1,·)$, $\chi_{660}(389,·)$, $\chi_{660}(391,·)$, $\chi_{660}(269,·)$, $\chi_{660}(271,·)$, $\chi_{660}(659,·)$, $\chi_{660}(149,·)$, $\chi_{660}(151,·)$, $\chi_{660}(541,·)$, $\chi_{660}(31,·)$, $\chi_{660}(419,·)$, $\chi_{660}(421,·)$, $\chi_{660}(299,·)$, $\chi_{660}(301,·)$, $\chi_{660}(29,·)$, $\chi_{660}(179,·)$, $\chi_{660}(181,·)$, $\chi_{660}(569,·)$, $\chi_{660}(59,·)$, $\chi_{660}(61,·)$, $\chi_{660}(449,·)$, $\chi_{660}(329,·)$, $\chi_{660}(331,·)$, $\chi_{660}(211,·)$, $\chi_{660}(599,·)$, $\chi_{660}(601,·)$, $\chi_{660}(91,·)$, $\chi_{660}(479,·)$, $\chi_{660}(481,·)$, $\chi_{660}(571,·)$, $\chi_{660}(359,·)$, $\chi_{660}(361,·)$, $\chi_{660}(239,·)$, $\chi_{660}(241,·)$, $\chi_{660}(629,·)$, $\chi_{660}(89,·)$, $\chi_{660}(631,·)$, $\chi_{660}(119,·)$, $\chi_{660}(509,·)$, $\chi_{660}(511,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{4} a^{21} - \frac{1}{4} a^{19} + \frac{1}{4} a^{17} - \frac{1}{4} a^{15} + \frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{22960624} a^{22} - \frac{7}{16} a^{20} - \frac{1}{16} a^{18} - \frac{7}{16} a^{16} - \frac{1}{16} a^{14} - \frac{7}{16} a^{12} - \frac{1}{16} a^{10} - \frac{7}{16} a^{8} - \frac{1}{16} a^{6} - \frac{7}{16} a^{4} - \frac{1}{16} a^{2} - \frac{510265}{1435039}$, $\frac{1}{91842496} a^{23} - \frac{7}{64} a^{21} + \frac{31}{64} a^{19} + \frac{9}{64} a^{17} + \frac{15}{64} a^{15} + \frac{25}{64} a^{13} - \frac{1}{64} a^{11} - \frac{23}{64} a^{9} - \frac{17}{64} a^{7} - \frac{7}{64} a^{5} + \frac{31}{64} a^{3} + \frac{2359813}{5740156} a$, $\frac{1}{367369984} a^{24} + \frac{7}{367369984} a^{22} + \frac{63}{256} a^{20} + \frac{41}{256} a^{18} + \frac{47}{256} a^{16} - \frac{71}{256} a^{14} + \frac{31}{256} a^{12} + \frac{73}{256} a^{10} + \frac{15}{256} a^{8} - \frac{39}{256} a^{6} - \frac{1}{256} a^{4} - \frac{510265}{22960624} a^{2} - \frac{133551}{1435039}$, $\frac{1}{1469479936} a^{25} + \frac{7}{1469479936} a^{23} + \frac{63}{1024} a^{21} + \frac{41}{1024} a^{19} + \frac{303}{1024} a^{17} + \frac{441}{1024} a^{15} + \frac{287}{1024} a^{13} + \frac{73}{1024} a^{11} + \frac{15}{1024} a^{9} - \frac{39}{1024} a^{7} + \frac{511}{1024} a^{5} + \frac{45410983}{91842496} a^{3} + \frac{2736527}{5740156} a$, $\frac{1}{5877919744} a^{26} + \frac{7}{5877919744} a^{24} + \frac{33}{5877919744} a^{22} + \frac{1577}{4096} a^{20} + \frac{1839}{4096} a^{18} - \frac{71}{4096} a^{16} - \frac{1249}{4096} a^{14} + \frac{585}{4096} a^{12} - \frac{497}{4096} a^{10} - \frac{551}{4096} a^{8} - \frac{1}{4096} a^{6} - \frac{510265}{367369984} a^{4} - \frac{133551}{22960624} a^{2} - \frac{26537}{1435039}$, $\frac{1}{23511678976} a^{27} + \frac{7}{23511678976} a^{25} + \frac{33}{23511678976} a^{23} + \frac{1577}{16384} a^{21} + \frac{1839}{16384} a^{19} + \frac{4025}{16384} a^{17} - \frac{1249}{16384} a^{15} - \frac{7607}{16384} a^{13} - \frac{497}{16384} a^{11} + \frac{3545}{16384} a^{9} - \frac{1}{16384} a^{7} - \frac{510265}{1469479936} a^{5} - \frac{133551}{91842496} a^{3} - \frac{26537}{5740156} a$, $\frac{1}{94046715904} a^{28} + \frac{7}{94046715904} a^{26} + \frac{33}{94046715904} a^{24} + \frac{119}{94046715904} a^{22} - \frac{30929}{65536} a^{20} + \frac{20409}{65536} a^{18} - \frac{17633}{65536} a^{16} + \frac{8777}{65536} a^{14} + \frac{15887}{65536} a^{12} - \frac{29223}{65536} a^{10} - \frac{1}{65536} a^{8} - \frac{510265}{5877919744} a^{6} - \frac{133551}{367369984} a^{4} - \frac{26537}{22960624} a^{2} - \frac{3263}{1435039}$, $\frac{1}{376186863616} a^{29} + \frac{7}{376186863616} a^{27} + \frac{33}{376186863616} a^{25} + \frac{119}{376186863616} a^{23} - \frac{30929}{262144} a^{21} + \frac{20409}{262144} a^{19} + \frac{113439}{262144} a^{17} - \frac{56759}{262144} a^{15} - \frac{115185}{262144} a^{13} + \frac{101849}{262144} a^{11} - \frac{65537}{262144} a^{9} + \frac{5877409479}{23511678976} a^{7} - \frac{367503535}{1469479936} a^{5} + \frac{22934087}{91842496} a^{3} - \frac{719151}{2870078} a$, $\frac{1}{1504747454464} a^{30} + \frac{7}{1504747454464} a^{28} + \frac{33}{1504747454464} a^{26} + \frac{119}{1504747454464} a^{24} + \frac{305}{1504747454464} a^{22} - \frac{45127}{1048576} a^{20} + \frac{178975}{1048576} a^{18} - \frac{122295}{1048576} a^{16} + \frac{474639}{1048576} a^{14} + \frac{36313}{1048576} a^{12} - \frac{1}{1048576} a^{10} - \frac{510265}{94046715904} a^{8} - \frac{133551}{5877919744} a^{6} - \frac{26537}{367369984} a^{4} - \frac{3263}{22960624} a^{2} + \frac{231}{1435039}$, $\frac{1}{6018989817856} a^{31} + \frac{7}{6018989817856} a^{29} + \frac{33}{6018989817856} a^{27} + \frac{119}{6018989817856} a^{25} + \frac{305}{6018989817856} a^{23} - \frac{45127}{4194304} a^{21} - \frac{869601}{4194304} a^{19} - \frac{1170871}{4194304} a^{17} + \frac{1523215}{4194304} a^{15} + \frac{36313}{4194304} a^{13} + \frac{1048575}{4194304} a^{11} - \frac{94047226169}{376186863616} a^{9} + \frac{5877786193}{23511678976} a^{7} - \frac{367396521}{1469479936} a^{5} + \frac{22957361}{91842496} a^{3} - \frac{358702}{1435039} a$, $\frac{1}{24075959271424} a^{32} + \frac{7}{24075959271424} a^{30} + \frac{33}{24075959271424} a^{28} + \frac{119}{24075959271424} a^{26} + \frac{305}{24075959271424} a^{24} + \frac{231}{24075959271424} a^{22} - \frac{6112481}{16777216} a^{20} + \frac{8266313}{16777216} a^{18} + \frac{4668943}{16777216} a^{16} + \frac{1084889}{16777216} a^{14} - \frac{1}{16777216} a^{12} - \frac{510265}{1504747454464} a^{10} - \frac{133551}{94046715904} a^{8} - \frac{26537}{5877919744} a^{6} - \frac{3263}{367369984} a^{4} + \frac{231}{22960624} a^{2} + \frac{305}{1435039}$, $\frac{1}{96303837085696} a^{33} + \frac{7}{96303837085696} a^{31} + \frac{33}{96303837085696} a^{29} + \frac{119}{96303837085696} a^{27} + \frac{305}{96303837085696} a^{25} + \frac{231}{96303837085696} a^{23} - \frac{6112481}{67108864} a^{21} + \frac{8266313}{67108864} a^{19} + \frac{21446159}{67108864} a^{17} + \frac{17862105}{67108864} a^{15} - \frac{16777217}{67108864} a^{13} + \frac{1504746944199}{6018989817856} a^{11} - \frac{94046849455}{376186863616} a^{9} + \frac{5877893207}{23511678976} a^{7} - \frac{367373247}{1469479936} a^{5} + \frac{22960855}{91842496} a^{3} - \frac{717367}{2870078} a$, $\frac{1}{385215348342784} a^{34} + \frac{7}{385215348342784} a^{32} + \frac{33}{385215348342784} a^{30} + \frac{119}{385215348342784} a^{28} + \frac{305}{385215348342784} a^{26} + \frac{231}{385215348342784} a^{24} - \frac{3263}{385215348342784} a^{22} + \frac{58597961}{268435456} a^{20} - \frac{28885489}{268435456} a^{18} - \frac{66023975}{268435456} a^{16} - \frac{1}{268435456} a^{14} - \frac{510265}{24075959271424} a^{12} - \frac{133551}{1504747454464} a^{10} - \frac{26537}{94046715904} a^{8} - \frac{3263}{5877919744} a^{6} + \frac{231}{367369984} a^{4} + \frac{305}{22960624} a^{2} + \frac{119}{1435039}$, $\frac{1}{1540861393371136} a^{35} + \frac{7}{1540861393371136} a^{33} + \frac{33}{1540861393371136} a^{31} + \frac{119}{1540861393371136} a^{29} + \frac{305}{1540861393371136} a^{27} + \frac{231}{1540861393371136} a^{25} - \frac{3263}{1540861393371136} a^{23} + \frac{58597961}{1073741824} a^{21} + \frac{239549967}{1073741824} a^{19} - \frac{334459431}{1073741824} a^{17} + \frac{268435455}{1073741824} a^{15} - \frac{24075959781689}{96303837085696} a^{13} + \frac{1504747320913}{6018989817856} a^{11} - \frac{94046742441}{376186863616} a^{9} + \frac{5877916481}{23511678976} a^{7} - \frac{367369753}{1469479936} a^{5} + \frac{22960929}{91842496} a^{3} - \frac{358730}{1435039} a$, $\frac{1}{6163445573484544} a^{36} + \frac{7}{6163445573484544} a^{34} + \frac{33}{6163445573484544} a^{32} + \frac{119}{6163445573484544} a^{30} + \frac{305}{6163445573484544} a^{28} + \frac{231}{6163445573484544} a^{26} - \frac{3263}{6163445573484544} a^{24} - \frac{26537}{6163445573484544} a^{22} - \frac{1102627313}{4294967296} a^{20} - \frac{66023975}{4294967296} a^{18} - \frac{1}{4294967296} a^{16} - \frac{510265}{385215348342784} a^{14} - \frac{133551}{24075959271424} a^{12} - \frac{26537}{1504747454464} a^{10} - \frac{3263}{94046715904} a^{8} + \frac{231}{5877919744} a^{6} + \frac{305}{367369984} a^{4} + \frac{119}{22960624} a^{2} + \frac{33}{1435039}$, $\frac{1}{24653782293938176} a^{37} + \frac{7}{24653782293938176} a^{35} + \frac{33}{24653782293938176} a^{33} + \frac{119}{24653782293938176} a^{31} + \frac{305}{24653782293938176} a^{29} + \frac{231}{24653782293938176} a^{27} - \frac{3263}{24653782293938176} a^{25} - \frac{26537}{24653782293938176} a^{23} - \frac{1102627313}{17179869184} a^{21} + \frac{4228943321}{17179869184} a^{19} - \frac{4294967297}{17179869184} a^{17} + \frac{385215347832519}{1540861393371136} a^{15} - \frac{24075959404975}{96303837085696} a^{13} + \frac{1504747427927}{6018989817856} a^{11} - \frac{94046719167}{376186863616} a^{9} + \frac{5877919975}{23511678976} a^{7} - \frac{367369679}{1469479936} a^{5} + \frac{22960743}{91842496} a^{3} - \frac{717503}{2870078} a$, $\frac{1}{98615129175752704} a^{38} + \frac{7}{98615129175752704} a^{36} + \frac{33}{98615129175752704} a^{34} + \frac{119}{98615129175752704} a^{32} + \frac{305}{98615129175752704} a^{30} + \frac{231}{98615129175752704} a^{28} - \frac{3263}{98615129175752704} a^{26} - \frac{26537}{98615129175752704} a^{24} - \frac{133551}{98615129175752704} a^{22} + \frac{17113845209}{68719476736} a^{20} - \frac{1}{68719476736} a^{18} - \frac{510265}{6163445573484544} a^{16} - \frac{133551}{385215348342784} a^{14} - \frac{26537}{24075959271424} a^{12} - \frac{3263}{1504747454464} a^{10} + \frac{231}{94046715904} a^{8} + \frac{305}{5877919744} a^{6} + \frac{119}{367369984} a^{4} + \frac{33}{22960624} a^{2} + \frac{7}{1435039}$, $\frac{1}{394460516703010816} a^{39} + \frac{7}{394460516703010816} a^{37} + \frac{33}{394460516703010816} a^{35} + \frac{119}{394460516703010816} a^{33} + \frac{305}{394460516703010816} a^{31} + \frac{231}{394460516703010816} a^{29} - \frac{3263}{394460516703010816} a^{27} - \frac{26537}{394460516703010816} a^{25} - \frac{133551}{394460516703010816} a^{23} + \frac{17113845209}{274877906944} a^{21} + \frac{68719476735}{274877906944} a^{19} - \frac{6163445573994809}{24653782293938176} a^{17} + \frac{385215348209233}{1540861393371136} a^{15} - \frac{24075959297961}{96303837085696} a^{13} + \frac{1504747451201}{6018989817856} a^{11} - \frac{94046715673}{376186863616} a^{9} + \frac{5877920049}{23511678976} a^{7} - \frac{367369865}{1469479936} a^{5} + \frac{22960657}{91842496} a^{3} - \frac{358758}{1435039} a$

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:  \( \frac{3}{91842496} a^{25} - \frac{17317525}{91842496} a^{3} \) (order $44$)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{165}) \), \(\Q(\sqrt{-165}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{15}) \), \(\Q(i, \sqrt{165})\), \(\Q(i, \sqrt{11})\), \(\Q(i, \sqrt{15})\), \(\Q(\sqrt{-11}, \sqrt{-15})\), \(\Q(\sqrt{11}, \sqrt{15})\), \(\Q(\sqrt{-11}, \sqrt{15})\), \(\Q(\sqrt{11}, \sqrt{-15})\), \(\Q(\zeta_{11})^+\), 8.0.189747360000.7, 10.0.219503494144.1, 10.10.1790566527853125.1, 10.0.1833540124521600000.1, \(\Q(\zeta_{11})\), \(\Q(\zeta_{44})^+\), 10.0.162778775259375.1, 10.10.166685465865600000.1, 20.0.3361869388230684433628866560000000000.3, \(\Q(\zeta_{44})\), 20.0.27784044530832102757263360000000000.3, 20.0.3206128490667995866421572265625.3, 20.20.3361869388230684433628866560000000000.2, 20.0.3361869388230684433628866560000000000.10, 20.0.3361869388230684433628866560000000000.5

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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed