Normalized defining polynomial
\( x^{32} - 119 x^{28} + 9880 x^{24} - 412321 x^{20} + 12533799 x^{16} - 204396521 x^{12} + 2295298360 x^{8} - 710952319 x^{4} + 214358881 \)
Invariants
| Degree: | $32$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 16]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1572926909253345615680132503044096000000000000000000000000=2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $61.29$ | 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: | $2, 3, 5, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(840=2^{3}\cdot 3\cdot 5\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(643,·)$, $\chi_{840}(517,·)$, $\chi_{840}(503,·)$, $\chi_{840}(13,·)$, $\chi_{840}(659,·)$, $\chi_{840}(281,·)$, $\chi_{840}(797,·)$, $\chi_{840}(799,·)$, $\chi_{840}(293,·)$, $\chi_{840}(167,·)$, $\chi_{840}(169,·)$, $\chi_{840}(29,·)$, $\chi_{840}(433,·)$, $\chi_{840}(307,·)$, $\chi_{840}(701,·)$, $\chi_{840}(449,·)$, $\chi_{840}(71,·)$, $\chi_{840}(713,·)$, $\chi_{840}(587,·)$, $\chi_{840}(589,·)$, $\chi_{840}(211,·)$, $\chi_{840}(727,·)$, $\chi_{840}(223,·)$, $\chi_{840}(97,·)$, $\chi_{840}(421,·)$, $\chi_{840}(491,·)$, $\chi_{840}(239,·)$, $\chi_{840}(83,·)$, $\chi_{840}(631,·)$, $\chi_{840}(377,·)$, $\chi_{840}(379,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{11} a^{17} - \frac{5}{11} a^{13} - \frac{1}{11} a^{9} + \frac{2}{11} a^{5} + \frac{5}{11} a$, $\frac{1}{11} a^{18} - \frac{5}{11} a^{14} - \frac{1}{11} a^{10} + \frac{2}{11} a^{6} + \frac{5}{11} a^{2}$, $\frac{1}{11} a^{19} - \frac{5}{11} a^{15} - \frac{1}{11} a^{11} + \frac{2}{11} a^{7} + \frac{5}{11} a^{3}$, $\frac{1}{11} a^{20} - \frac{5}{11} a^{16} - \frac{1}{11} a^{12} + \frac{2}{11} a^{8} + \frac{5}{11} a^{4}$, $\frac{1}{11} a^{21} - \frac{4}{11} a^{13} - \frac{3}{11} a^{9} + \frac{4}{11} a^{5} + \frac{3}{11} a$, $\frac{1}{11} a^{22} - \frac{4}{11} a^{14} - \frac{3}{11} a^{10} + \frac{4}{11} a^{6} + \frac{3}{11} a^{2}$, $\frac{1}{11} a^{23} - \frac{4}{11} a^{15} - \frac{3}{11} a^{11} + \frac{4}{11} a^{7} + \frac{3}{11} a^{3}$, $\frac{1}{5718889} a^{24} - \frac{82223}{5718889} a^{20} - \frac{1137623}{5718889} a^{16} + \frac{489429}{5718889} a^{12} + \frac{2571269}{5718889} a^{8} + \frac{414691}{5718889} a^{4} - \frac{250180}{519899}$, $\frac{1}{5718889} a^{25} - \frac{82223}{5718889} a^{21} - \frac{97825}{5718889} a^{17} + \frac{1009328}{5718889} a^{13} + \frac{1531471}{5718889} a^{9} + \frac{2494287}{5718889} a^{5} + \frac{2447010}{5718889} a$, $\frac{1}{62907779} a^{26} - \frac{82223}{62907779} a^{22} - \frac{1657522}{62907779} a^{18} + \frac{25964480}{62907779} a^{14} + \frac{14528946}{62907779} a^{10} + \frac{22250449}{62907779} a^{6} - \frac{11070364}{62907779} a^{2}$, $\frac{1}{691985569} a^{27} + \frac{5636666}{691985569} a^{23} + \frac{21218034}{691985569} a^{19} + \frac{140342260}{691985569} a^{15} - \frac{277134393}{691985569} a^{11} + \frac{153784896}{691985569} a^{7} - \frac{319890370}{691985569} a^{3}$, $\frac{1}{1717716835846477797220857563101} a^{28} + \frac{114033223912848374268169}{1717716835846477797220857563101} a^{24} + \frac{77892654761026552227194238453}{1717716835846477797220857563101} a^{20} - \frac{24722545893826316678534519521}{59231615029188889559339915969} a^{16} + \frac{721622702604598772677620617865}{1717716835846477797220857563101} a^{12} + \frac{589997015450983266684979676443}{1717716835846477797220857563101} a^{8} - \frac{302278063176266036316919213991}{1717716835846477797220857563101} a^{4} - \frac{367243436920092426856531}{1485093305557289523542659}$, $\frac{1}{18894885194311255769429433194111} a^{29} - \frac{787042231046923251245558}{18894885194311255769429433194111} a^{25} - \frac{4174294091859581519177637617}{18894885194311255769429433194111} a^{21} - \frac{21682969811707707738331749046}{651547765321077785152739075659} a^{17} - \frac{1280950516101341655335807785928}{18894885194311255769429433194111} a^{13} - \frac{7192373028059494024962622424805}{18894885194311255769429433194111} a^{9} + \frac{1978707347093749722413947692899}{18894885194311255769429433194111} a^{5} + \frac{257679829571125697062715172}{1290546082529284595958570671} a$, $\frac{1}{207843737137423813463723765135221} a^{30} - \frac{787042231046923251245558}{207843737137423813463723765135221} a^{26} + \frac{5148976213447573810143395051686}{207843737137423813463723765135221} a^{22} + \frac{37548645217481181821008166923}{7167025418531855636680129832249} a^{18} - \frac{87166792308425231516378685940978}{207843737137423813463723765135221} a^{14} - \frac{43264426580835527766600631249926}{207843737137423813463723765135221} a^{10} + \frac{101606283826189461961223686352757}{207843737137423813463723765135221} a^{6} + \frac{47701084901697038826252959}{117322371139025872359870061} a^{2}$, $\frac{1}{2286281108511661948100961416487431} a^{31} - \frac{787042231046923251245558}{2286281108511661948100961416487431} a^{27} + \frac{5148976213447573810143395051686}{2286281108511661948100961416487431} a^{23} + \frac{1992191941180714537279225393900}{78837279603850412003481428154739} a^{19} - \frac{370590070223094068057820183852643}{2286281108511661948100961416487431} a^{15} - \frac{99949082163769295074888930832259}{2286281108511661948100961416487431} a^{11} + \frac{1046350543541752250432695346058307}{2286281108511661948100961416487431} a^{7} - \frac{4168182911642367467306019891}{14196006907822130555544277381} a^{3}$
Class group and class number
Not computed
Unit group
| Rank: | $15$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{8184204619242290912444619}{2286281108511661948100961416487431} a^{31} + \frac{971534876001797854027658680}{2286281108511661948100961416487431} a^{27} - \frac{80573107729556293214766066251}{2286281108511661948100961416487431} a^{23} + \frac{115539795677410049688995318391}{78837279603850412003481428154739} a^{19} - \frac{101570313606890998988757203748680}{2286281108511661948100961416487431} a^{15} + \frac{1642033186349824494341346652528880}{2286281108511661948100961416487431} a^{11} - \frac{18270397561013005214299301356543459}{2286281108511661948100961416487431} a^{7} + \frac{7937456859927802581597201}{117322371139025872359870061} a^{3} \) (order $24$) | 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
$C_2^3\times C_4$ (as 32T34):
| An abelian group of order 32 |
| The 32 conjugacy class representatives for $C_2^3\times C_4$ |
| Character table for $C_2^3\times C_4$ is not computed |
Intermediate fields
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 | R | ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $7$ | 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |