↑

Properties

Label 40.0.64595689514...1936.1
Degree $40$
Signature $[0, 20]$
Discriminant $2^{40}\cdot 11^{36}\cdot 13^{20}$
Root discriminant $62.41$
Ramified primes $2, 11, 13$
Class number Not computed
Class group Not computed
Galois group $C_2^2\times C_{10}$ (as 40T7)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3486784401, 0, -2711943423, 0, 1721868840, 0, -1037904273, 0, 615940119, 0, -363741840, 0, 214472529, 0, -126396207, 0, 74477880, 0, -43883217, 0, 25856071, 0, -4875913, 0, 919480, 0, -173383, 0, 32689, 0, -6160, 0, 1159, 0, -217, 0, 40, 0, -7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 7*x^38 + 40*x^36 - 217*x^34 + 1159*x^32 - 6160*x^30 + 32689*x^28 - 173383*x^26 + 919480*x^24 - 4875913*x^22 + 25856071*x^20 - 43883217*x^18 + 74477880*x^16 - 126396207*x^14 + 214472529*x^12 - 363741840*x^10 + 615940119*x^8 - 1037904273*x^6 + 1721868840*x^4 - 2711943423*x^2 + 3486784401)
 
gp: K = bnfinit(x^40 - 7*x^38 + 40*x^36 - 217*x^34 + 1159*x^32 - 6160*x^30 + 32689*x^28 - 173383*x^26 + 919480*x^24 - 4875913*x^22 + 25856071*x^20 - 43883217*x^18 + 74477880*x^16 - 126396207*x^14 + 214472529*x^12 - 363741840*x^10 + 615940119*x^8 - 1037904273*x^6 + 1721868840*x^4 - 2711943423*x^2 + 3486784401, 1)
 

Normalized defining polynomial

\( x^{40} - 7 x^{38} + 40 x^{36} - 217 x^{34} + 1159 x^{32} - 6160 x^{30} + 32689 x^{28} - 173383 x^{26} + 919480 x^{24} - 4875913 x^{22} + 25856071 x^{20} - 43883217 x^{18} + 74477880 x^{16} - 126396207 x^{14} + 214472529 x^{12} - 363741840 x^{10} + 615940119 x^{8} - 1037904273 x^{6} + 1721868840 x^{4} - 2711943423 x^{2} + 3486784401 \)

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:  \(645956895148959968043416875848059904118905976531208178355650775729831936=2^{40}\cdot 11^{36}\cdot 13^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.41$
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:  $2, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(572=2^{2}\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{572}(1,·)$, $\chi_{572}(259,·)$, $\chi_{572}(389,·)$, $\chi_{572}(519,·)$, $\chi_{572}(521,·)$, $\chi_{572}(365,·)$, $\chi_{572}(131,·)$, $\chi_{572}(25,·)$, $\chi_{572}(27,·)$, $\chi_{572}(157,·)$, $\chi_{572}(415,·)$, $\chi_{572}(545,·)$, $\chi_{572}(547,·)$, $\chi_{572}(261,·)$, $\chi_{572}(129,·)$, $\chi_{572}(391,·)$, $\chi_{572}(285,·)$, $\chi_{572}(51,·)$, $\chi_{572}(155,·)$, $\chi_{572}(181,·)$, $\chi_{572}(311,·)$, $\chi_{572}(313,·)$, $\chi_{572}(287,·)$, $\chi_{572}(53,·)$, $\chi_{572}(417,·)$, $\chi_{572}(183,·)$, $\chi_{572}(79,·)$, $\chi_{572}(337,·)$, $\chi_{572}(339,·)$, $\chi_{572}(469,·)$, $\chi_{572}(441,·)$, $\chi_{572}(207,·)$, $\chi_{572}(571,·)$, $\chi_{572}(103,·)$, $\chi_{572}(233,·)$, $\chi_{572}(235,·)$, $\chi_{572}(493,·)$, $\chi_{572}(467,·)$, $\chi_{572}(105,·)$, $\chi_{572}(443,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{3} a^{21} - \frac{1}{3} a^{19} + \frac{1}{3} a^{17} - \frac{1}{3} a^{15} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{232704639} a^{22} - \frac{1}{9} a^{20} - \frac{2}{9} a^{18} - \frac{4}{9} a^{16} + \frac{1}{9} a^{14} + \frac{2}{9} a^{12} + \frac{4}{9} a^{10} - \frac{1}{9} a^{8} - \frac{2}{9} a^{6} - \frac{4}{9} a^{4} + \frac{1}{9} a^{2} - \frac{4875913}{25856071}$, $\frac{1}{698113917} a^{23} - \frac{1}{27} a^{21} - \frac{2}{27} a^{19} - \frac{4}{27} a^{17} - \frac{8}{27} a^{15} + \frac{11}{27} a^{13} - \frac{5}{27} a^{11} - \frac{10}{27} a^{9} + \frac{7}{27} a^{7} - \frac{13}{27} a^{5} + \frac{1}{27} a^{3} - \frac{4875913}{77568213} a$, $\frac{1}{2094341751} a^{24} + \frac{2}{2094341751} a^{22} - \frac{11}{81} a^{20} + \frac{5}{81} a^{18} - \frac{17}{81} a^{16} - \frac{7}{81} a^{14} + \frac{40}{81} a^{12} + \frac{26}{81} a^{10} + \frac{25}{81} a^{8} - \frac{4}{81} a^{6} - \frac{35}{81} a^{4} + \frac{6993386}{77568213} a^{2} - \frac{3956433}{25856071}$, $\frac{1}{6283025253} a^{25} + \frac{2}{6283025253} a^{23} - \frac{11}{243} a^{21} + \frac{5}{243} a^{19} + \frac{64}{243} a^{17} - \frac{7}{243} a^{15} - \frac{41}{243} a^{13} + \frac{107}{243} a^{11} + \frac{106}{243} a^{9} - \frac{4}{243} a^{7} + \frac{46}{243} a^{5} + \frac{84561599}{232704639} a^{3} - \frac{29812504}{77568213} a$, $\frac{1}{18849075759} a^{26} + \frac{2}{18849075759} a^{24} - \frac{1}{819525033} a^{22} - \frac{319}{729} a^{20} + \frac{145}{729} a^{18} - \frac{331}{729} a^{16} + \frac{283}{729} a^{14} + \frac{269}{729} a^{12} - \frac{56}{729} a^{10} + \frac{158}{729} a^{8} + \frac{127}{729} a^{6} + \frac{6993386}{698113917} a^{4} - \frac{1318811}{77568213} a^{2} + \frac{32439}{1124177}$, $\frac{1}{56547227277} a^{27} + \frac{2}{56547227277} a^{25} - \frac{1}{2458575099} a^{23} - \frac{319}{2187} a^{21} - \frac{584}{2187} a^{19} + \frac{398}{2187} a^{17} + \frac{283}{2187} a^{15} + \frac{998}{2187} a^{13} - \frac{785}{2187} a^{11} + \frac{887}{2187} a^{9} + \frac{856}{2187} a^{7} + \frac{6993386}{2094341751} a^{5} - \frac{1318811}{232704639} a^{3} + \frac{10813}{1124177} a$, $\frac{1}{169641681831} a^{28} + \frac{2}{169641681831} a^{26} - \frac{1}{7375725297} a^{24} + \frac{143}{169641681831} a^{22} - \frac{584}{6561} a^{20} + \frac{398}{6561} a^{18} + \frac{2470}{6561} a^{16} - \frac{1189}{6561} a^{14} - \frac{785}{6561} a^{12} + \frac{3074}{6561} a^{10} - \frac{1331}{6561} a^{8} + \frac{6993386}{6283025253} a^{6} - \frac{1318811}{698113917} a^{4} + \frac{10813}{3372531} a^{2} - \frac{140694}{25856071}$, $\frac{1}{508925045493} a^{29} + \frac{2}{508925045493} a^{27} - \frac{1}{22127175891} a^{25} + \frac{143}{508925045493} a^{23} - \frac{584}{19683} a^{21} + \frac{6959}{19683} a^{19} - \frac{4091}{19683} a^{17} + \frac{5372}{19683} a^{15} - \frac{785}{19683} a^{13} - \frac{3487}{19683} a^{11} - \frac{7892}{19683} a^{9} - \frac{6276031867}{18849075759} a^{7} + \frac{696795106}{2094341751} a^{5} - \frac{3361718}{10117593} a^{3} + \frac{25715377}{77568213} a$, $\frac{1}{1526775136479} a^{30} + \frac{2}{1526775136479} a^{28} - \frac{1}{66381527673} a^{26} + \frac{143}{1526775136479} a^{24} - \frac{794}{1526775136479} a^{22} - \frac{25846}{59049} a^{20} + \frac{9031}{59049} a^{18} - \frac{7750}{59049} a^{16} - \frac{27029}{59049} a^{14} + \frac{22757}{59049} a^{12} + \frac{24913}{59049} a^{10} + \frac{6993386}{56547227277} a^{8} - \frac{1318811}{6283025253} a^{6} + \frac{10813}{30352779} a^{4} - \frac{46898}{77568213} a^{2} + \frac{26529}{25856071}$, $\frac{1}{4580325409437} a^{31} + \frac{2}{4580325409437} a^{29} - \frac{1}{199144583019} a^{27} + \frac{143}{4580325409437} a^{25} - \frac{794}{4580325409437} a^{23} - \frac{25846}{177147} a^{21} + \frac{68080}{177147} a^{19} - \frac{66799}{177147} a^{17} + \frac{32020}{177147} a^{15} + \frac{22757}{177147} a^{13} + \frac{83962}{177147} a^{11} + \frac{56554220663}{169641681831} a^{9} - \frac{6284344064}{18849075759} a^{7} + \frac{30363592}{91058337} a^{5} - \frac{77615111}{232704639} a^{3} + \frac{25882600}{77568213} a$, $\frac{1}{13740976228311} a^{32} + \frac{2}{13740976228311} a^{30} - \frac{1}{597433749057} a^{28} + \frac{143}{13740976228311} a^{26} - \frac{794}{13740976228311} a^{24} + \frac{4271}{13740976228311} a^{22} - \frac{227165}{531441} a^{20} + \frac{228446}{531441} a^{18} - \frac{86078}{531441} a^{16} + \frac{140855}{531441} a^{14} - \frac{211283}{531441} a^{12} + \frac{6993386}{508925045493} a^{10} - \frac{1318811}{56547227277} a^{8} + \frac{10813}{273175011} a^{6} - \frac{46898}{698113917} a^{4} + \frac{8843}{77568213} a^{2} - \frac{5001}{25856071}$, $\frac{1}{41222928684933} a^{33} + \frac{2}{41222928684933} a^{31} - \frac{1}{1792301247171} a^{29} + \frac{143}{41222928684933} a^{27} - \frac{794}{41222928684933} a^{25} + \frac{4271}{41222928684933} a^{23} - \frac{227165}{1594323} a^{21} + \frac{759887}{1594323} a^{19} - \frac{86078}{1594323} a^{17} + \frac{140855}{1594323} a^{15} - \frac{211283}{1594323} a^{13} + \frac{6993386}{1526775136479} a^{11} - \frac{1318811}{169641681831} a^{9} + \frac{10813}{819525033} a^{7} - \frac{46898}{2094341751} a^{5} + \frac{8843}{232704639} a^{3} - \frac{1667}{25856071} a$, $\frac{1}{123668786054799} a^{34} + \frac{2}{123668786054799} a^{32} - \frac{1}{5376903741513} a^{30} + \frac{143}{123668786054799} a^{28} - \frac{794}{123668786054799} a^{26} + \frac{4271}{123668786054799} a^{24} - \frac{22751}{123668786054799} a^{22} - \frac{834436}{4782969} a^{20} - \frac{1680401}{4782969} a^{18} + \frac{140855}{4782969} a^{16} - \frac{211283}{4782969} a^{14} + \frac{6993386}{4580325409437} a^{12} - \frac{1318811}{508925045493} a^{10} + \frac{10813}{2458575099} a^{8} - \frac{46898}{6283025253} a^{6} + \frac{8843}{698113917} a^{4} - \frac{1667}{77568213} a^{2} + \frac{942}{25856071}$, $\frac{1}{371006358164397} a^{35} + \frac{2}{371006358164397} a^{33} - \frac{1}{16130711224539} a^{31} + \frac{143}{371006358164397} a^{29} - \frac{794}{371006358164397} a^{27} + \frac{4271}{371006358164397} a^{25} - \frac{22751}{371006358164397} a^{23} - \frac{834436}{14348907} a^{21} - \frac{6463370}{14348907} a^{19} - \frac{4642114}{14348907} a^{17} + \frac{4571686}{14348907} a^{15} - \frac{4580318416051}{13740976228311} a^{13} + \frac{508923726682}{1526775136479} a^{11} - \frac{2458564286}{7375725297} a^{9} + \frac{6282978355}{18849075759} a^{7} - \frac{698105074}{2094341751} a^{5} + \frac{77566546}{232704639} a^{3} - \frac{25855129}{77568213} a$, $\frac{1}{1113019074493191} a^{36} + \frac{2}{1113019074493191} a^{34} - \frac{1}{48392133673617} a^{32} + \frac{143}{1113019074493191} a^{30} - \frac{794}{1113019074493191} a^{28} + \frac{4271}{1113019074493191} a^{26} - \frac{22751}{1113019074493191} a^{24} + \frac{120818}{1113019074493191} a^{22} + \frac{17451475}{43046721} a^{20} + \frac{14489762}{43046721} a^{18} - \frac{211283}{43046721} a^{16} + \frac{6993386}{41222928684933} a^{14} - \frac{1318811}{4580325409437} a^{12} + \frac{10813}{22127175891} a^{10} - \frac{46898}{56547227277} a^{8} + \frac{8843}{6283025253} a^{6} - \frac{1667}{698113917} a^{4} + \frac{314}{77568213} a^{2} - \frac{177}{25856071}$, $\frac{1}{3339057223479573} a^{37} + \frac{2}{3339057223479573} a^{35} - \frac{1}{145176401020851} a^{33} + \frac{143}{3339057223479573} a^{31} - \frac{794}{3339057223479573} a^{29} + \frac{4271}{3339057223479573} a^{27} - \frac{22751}{3339057223479573} a^{25} + \frac{120818}{3339057223479573} a^{23} + \frac{17451475}{129140163} a^{21} - \frac{28556959}{129140163} a^{19} + \frac{42835438}{129140163} a^{17} + \frac{6993386}{123668786054799} a^{15} - \frac{1318811}{13740976228311} a^{13} + \frac{10813}{66381527673} a^{11} - \frac{46898}{169641681831} a^{9} + \frac{8843}{18849075759} a^{7} - \frac{1667}{2094341751} a^{5} + \frac{314}{232704639} a^{3} - \frac{59}{25856071} a$, $\frac{1}{10017171670438719} a^{38} + \frac{2}{10017171670438719} a^{36} - \frac{1}{435529203062553} a^{34} + \frac{143}{10017171670438719} a^{32} - \frac{794}{10017171670438719} a^{30} + \frac{4271}{10017171670438719} a^{28} - \frac{22751}{10017171670438719} a^{26} + \frac{120818}{10017171670438719} a^{24} - \frac{640967}{10017171670438719} a^{22} - \frac{28556959}{387420489} a^{20} + \frac{42835438}{387420489} a^{18} + \frac{6993386}{371006358164397} a^{16} - \frac{1318811}{41222928684933} a^{14} + \frac{10813}{199144583019} a^{12} - \frac{46898}{508925045493} a^{10} + \frac{8843}{56547227277} a^{8} - \frac{1667}{6283025253} a^{6} + \frac{314}{698113917} a^{4} - \frac{59}{77568213} a^{2} + \frac{33}{25856071}$, $\frac{1}{30051515011316157} a^{39} + \frac{2}{30051515011316157} a^{37} - \frac{1}{1306587609187659} a^{35} + \frac{143}{30051515011316157} a^{33} - \frac{794}{30051515011316157} a^{31} + \frac{4271}{30051515011316157} a^{29} - \frac{22751}{30051515011316157} a^{27} + \frac{120818}{30051515011316157} a^{25} - \frac{640967}{30051515011316157} a^{23} - \frac{28556959}{1162261467} a^{21} + \frac{42835438}{1162261467} a^{19} + \frac{6993386}{1113019074493191} a^{17} - \frac{1318811}{123668786054799} a^{15} + \frac{10813}{597433749057} a^{13} - \frac{46898}{1526775136479} a^{11} + \frac{8843}{169641681831} a^{9} - \frac{1667}{18849075759} a^{7} + \frac{314}{2094341751} a^{5} - \frac{59}{232704639} a^{3} + \frac{11}{25856071} a$

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:  \( -\frac{75316}{371006358164397} a^{37} - \frac{7020022316929}{371006358164397} a^{15} \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-143}) \), \(\Q(\sqrt{143}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-11}) \), \(\Q(i, \sqrt{13})\), \(\Q(i, \sqrt{143})\), \(\Q(i, \sqrt{11})\), \(\Q(\sqrt{11}, \sqrt{-13})\), \(\Q(\sqrt{-11}, \sqrt{-13})\), \(\Q(\sqrt{-11}, \sqrt{13})\), \(\Q(\sqrt{11}, \sqrt{13})\), \(\Q(\zeta_{11})^+\), 8.0.107049369856.1, 10.0.219503494144.1, 10.0.81500110851208192.3, 10.10.79589952003133.1, 10.0.875489472034463.1, 10.10.896501219363290112.1, \(\Q(\zeta_{44})^+\), \(\Q(\zeta_{11})\), 20.0.6642268068759223286357626127908864.1, 20.0.803714436319866017649272761476972544.4, \(\Q(\zeta_{44})\), 20.0.803714436319866017649272761476972544.2, 20.0.803714436319866017649272761476972544.5, 20.0.766481815643182771348259698369.1, 20.20.803714436319866017649272761476972544.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ R R ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
11Data not computed
13Data not computed