Normalized defining polynomial
\( x^{28} + 429 x^{24} + 31802 x^{20} + 705923 x^{16} + 6451930 x^{12} + 25926230 x^{8} + 39023453 x^{4} + 5764801 \)
Invariants
| Degree: | $28$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 14]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(115059555281167207950880893411598858797306743145141108736=2^{56}\cdot 43^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $100.50$ | 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: | $2, 43$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(344=2^{3}\cdot 43\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{344}(1,·)$, $\chi_{344}(259,·)$, $\chi_{344}(133,·)$, $\chi_{344}(193,·)$, $\chi_{344}(11,·)$, $\chi_{344}(269,·)$, $\chi_{344}(207,·)$, $\chi_{344}(145,·)$, $\chi_{344}(107,·)$, $\chi_{344}(21,·)$, $\chi_{344}(279,·)$, $\chi_{344}(127,·)$, $\chi_{344}(219,·)$, $\chi_{344}(97,·)$, $\chi_{344}(35,·)$, $\chi_{344}(293,·)$, $\chi_{344}(231,·)$, $\chi_{344}(41,·)$, $\chi_{344}(299,·)$, $\chi_{344}(173,·)$, $\chi_{344}(47,·)$, $\chi_{344}(305,·)$, $\chi_{344}(183,·)$, $\chi_{344}(121,·)$, $\chi_{344}(87,·)$, $\chi_{344}(59,·)$, $\chi_{344}(317,·)$, $\chi_{344}(213,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} + \frac{1}{7} a$, $\frac{1}{7} a^{14} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{3}$, $\frac{1}{259} a^{16} + \frac{18}{37} a^{12} + \frac{15}{37} a^{8} - \frac{13}{259} a^{4} + \frac{1}{37}$, $\frac{1}{259} a^{17} + \frac{15}{259} a^{13} + \frac{15}{37} a^{9} - \frac{13}{259} a^{5} - \frac{104}{259} a$, $\frac{1}{1813} a^{18} + \frac{15}{1813} a^{14} - \frac{96}{259} a^{10} + \frac{505}{1813} a^{6} - \frac{622}{1813} a^{2}$, $\frac{1}{12691} a^{19} + \frac{792}{12691} a^{15} + \frac{681}{1813} a^{11} + \frac{4131}{12691} a^{7} - \frac{1658}{12691} a^{3}$, $\frac{1}{12691} a^{20} + \frac{8}{12691} a^{16} - \frac{20}{49} a^{12} - \frac{2043}{12691} a^{8} - \frac{4157}{12691} a^{4} - \frac{16}{37}$, $\frac{1}{12691} a^{21} + \frac{8}{12691} a^{17} + \frac{1}{49} a^{13} - \frac{2043}{12691} a^{9} - \frac{4157}{12691} a^{5} - \frac{1}{259} a$, $\frac{1}{12691} a^{22} + \frac{1}{12691} a^{18} + \frac{22}{1813} a^{14} + \frac{2661}{12691} a^{10} + \frac{4999}{12691} a^{6} + \frac{615}{1813} a^{2}$, $\frac{1}{12691} a^{23} - \frac{638}{12691} a^{15} - \frac{2106}{12691} a^{11} + \frac{124}{1813} a^{7} + \frac{5963}{12691} a^{3}$, $\frac{1}{17104636506372544633} a^{24} - \frac{640857574150393}{17104636506372544633} a^{20} + \frac{16503037258606480}{17104636506372544633} a^{16} - \frac{7197273607890289111}{17104636506372544633} a^{12} - \frac{8356086519836747410}{17104636506372544633} a^{8} + \frac{8248195887662146200}{17104636506372544633} a^{4} - \frac{2728072365356126}{7123963559505433}$, $\frac{1}{119732455544607812431} a^{25} - \frac{640857574150393}{119732455544607812431} a^{21} + \frac{16503037258606480}{119732455544607812431} a^{17} - \frac{7197273607890289111}{119732455544607812431} a^{13} - \frac{8356086519836747410}{119732455544607812431} a^{9} + \frac{42457468900407235466}{119732455544607812431} a^{5} - \frac{2728072365356126}{49867744916538031} a$, $\frac{1}{838127188812254687017} a^{26} - \frac{28944172256509816}{838127188812254687017} a^{22} - \frac{209923480200268904}{838127188812254687017} a^{18} - \frac{48737105123366468934}{838127188812254687017} a^{14} - \frac{309729781257599883514}{838127188812254687017} a^{10} + \frac{160114348034975356877}{838127188812254687017} a^{6} - \frac{23714516509570594}{49867744916538031} a^{2}$, $\frac{1}{5866890321685782809119} a^{27} - \frac{94985239848681803}{5866890321685782809119} a^{23} + \frac{186322925352763018}{5866890321685782809119} a^{19} + \frac{187491793653832728565}{5866890321685782809119} a^{15} + \frac{880132133550562806265}{5866890321685782809119} a^{11} + \frac{2577811791516799628960}{5866890321685782809119} a^{7} - \frac{304437556088194329}{2443519500910363519} a^{3}$
Class group and class number
$C_{8729}$, which has order $8729$ (assuming GRH)
Unit group
| Rank: | $13$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{41365516137009}{838127188812254687017} a^{27} - \frac{18293808923362121}{838127188812254687017} a^{23} - \frac{1546463035435128399}{838127188812254687017} a^{19} - \frac{44882858525823238397}{838127188812254687017} a^{15} - \frac{534364444835731415831}{838127188812254687017} a^{11} - \frac{2519906511884383877690}{838127188812254687017} a^{7} - \frac{10605018042069296065}{2443519500910363519} a^{3} \) (order $8$) | 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: | \( 9818745262125.305 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_{14}$ (as 28T2):
| An abelian group of order 28 |
| The 28 conjugacy class representatives for $C_2\times C_{14}$ |
| Character table for $C_2\times C_{14}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{8})\), 7.7.6321363049.1, 14.14.83801419645740806624509952.1, 14.0.654698590982350051753984.1, 14.0.83801419645740806624509952.1 |
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 | ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{14}$ | ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/17.7.0.1}{7} }^{4}$ | ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{14}$ | ${\href{/LocalNumberField/41.7.0.1}{7} }^{4}$ | R | ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/59.14.0.1}{14} }^{2}$ |
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 | ||||||
| $43$ | 43.14.12.1 | $x^{14} + 3569 x^{7} + 4043763$ | $7$ | $2$ | $12$ | $C_{14}$ | $[\ ]_{7}^{2}$ |
| 43.14.12.1 | $x^{14} + 3569 x^{7} + 4043763$ | $7$ | $2$ | $12$ | $C_{14}$ | $[\ ]_{7}^{2}$ | |