Normalized defining polynomial
\( x^{28} - 12 x^{27} + 97 x^{26} - 616 x^{25} + 3227 x^{24} - 14252 x^{23} + 54685 x^{22} - 183020 x^{21} + 539214 x^{20} - 1401648 x^{19} + 3232728 x^{18} - 6601344 x^{17} + 11989926 x^{16} - 19294392 x^{15} + 28024356 x^{14} - 36414037 x^{13} + 44459150 x^{12} - 50087517 x^{11} + 54111340 x^{10} - 47799819 x^{9} + 31974029 x^{8} - 11720297 x^{7} + 14313843 x^{6} - 10672203 x^{5} + 8855016 x^{4} + 6890309 x^{3} + 6869672 x^{2} + 1932120 x + 490421 \)
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: | \(2002407232668075518060451991052904113005433477935650601=7^{14}\cdot 43^{26}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $86.96$ | 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: | $7, 43$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(301=7\cdot 43\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{301}(64,·)$, $\chi_{301}(1,·)$, $\chi_{301}(260,·)$, $\chi_{301}(8,·)$, $\chi_{301}(204,·)$, $\chi_{301}(78,·)$, $\chi_{301}(274,·)$, $\chi_{301}(211,·)$, $\chi_{301}(85,·)$, $\chi_{301}(22,·)$, $\chi_{301}(279,·)$, $\chi_{301}(216,·)$, $\chi_{301}(90,·)$, $\chi_{301}(27,·)$, $\chi_{301}(223,·)$, $\chi_{301}(97,·)$, $\chi_{301}(293,·)$, $\chi_{301}(41,·)$, $\chi_{301}(300,·)$, $\chi_{301}(237,·)$, $\chi_{301}(174,·)$, $\chi_{301}(176,·)$, $\chi_{301}(113,·)$, $\chi_{301}(118,·)$, $\chi_{301}(183,·)$, $\chi_{301}(188,·)$, $\chi_{301}(125,·)$, $\chi_{301}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{27501559} a^{25} + \frac{3938535}{27501559} a^{24} + \frac{11211562}{27501559} a^{23} - \frac{8794301}{27501559} a^{22} + \frac{2035927}{27501559} a^{21} + \frac{3884138}{27501559} a^{20} + \frac{1837291}{27501559} a^{19} + \frac{8949419}{27501559} a^{18} - \frac{4598374}{27501559} a^{17} - \frac{1321352}{27501559} a^{16} + \frac{3454693}{27501559} a^{15} + \frac{10002647}{27501559} a^{14} + \frac{2991626}{27501559} a^{13} - \frac{2011914}{27501559} a^{12} + \frac{10962408}{27501559} a^{11} + \frac{7945669}{27501559} a^{10} - \frac{5923078}{27501559} a^{9} + \frac{5190586}{27501559} a^{8} - \frac{9416517}{27501559} a^{7} - \frac{8197124}{27501559} a^{6} - \frac{7144380}{27501559} a^{5} - \frac{6311610}{27501559} a^{4} - \frac{2730071}{27501559} a^{3} - \frac{10379014}{27501559} a^{2} - \frac{9265738}{27501559} a + \frac{75985}{27501559}$, $\frac{1}{1744091421023249} a^{26} + \frac{21363943}{1744091421023249} a^{25} + \frac{303601985495712}{1744091421023249} a^{24} - \frac{276714425280256}{1744091421023249} a^{23} - \frac{559269102575299}{1744091421023249} a^{22} - \frac{28091808544109}{1744091421023249} a^{21} + \frac{163669072462339}{1744091421023249} a^{20} - \frac{612858904707762}{1744091421023249} a^{19} + \frac{39684239656165}{1744091421023249} a^{18} - \frac{561315814305337}{1744091421023249} a^{17} + \frac{127160946943636}{1744091421023249} a^{16} + \frac{248090096130391}{1744091421023249} a^{15} + \frac{381862971801333}{1744091421023249} a^{14} - \frac{257935310283472}{1744091421023249} a^{13} + \frac{799400217188066}{1744091421023249} a^{12} + \frac{245776633783817}{1744091421023249} a^{11} + \frac{439800093207667}{1744091421023249} a^{10} + \frac{578674423997256}{1744091421023249} a^{9} - \frac{785111529063263}{1744091421023249} a^{8} + \frac{643714653533692}{1744091421023249} a^{7} + \frac{525658658277428}{1744091421023249} a^{6} + \frac{857436798160769}{1744091421023249} a^{5} - \frac{663912650647294}{1744091421023249} a^{4} - \frac{84768854992308}{1744091421023249} a^{3} + \frac{835454069567502}{1744091421023249} a^{2} + \frac{288119775604599}{1744091421023249} a - \frac{574057552431258}{1744091421023249}$, $\frac{1}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{27} + \frac{12376981090415413791176281941092827101220315714941118013319653491853301372}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{26} + \frac{427226458498973840089920226779709235137608362981792315236095806103536041432335805}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{25} + \frac{65713806382915731245321014882877251496080689368950532942320094667233704440676896659450251}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{24} + \frac{76694017252678703642362976727985235156774587951543148995007880721482075467245684223367372}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{23} - \frac{102716352583946056490772552793416219051002775311318617018330630999425594575707224587959058}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{22} + \frac{34579999538945460786019945590608233625553044616758307237880469314985088455612583573519769}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{21} + \frac{106633991253977687833891479219622954133969492565418490319115239638996413584090785062871196}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{20} + \frac{6025509588750713099254917086871010967031228414908824261016394148439757041277734385872161}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{19} - \frac{53361899560384829346018795663100319884760955128422850517200282578460389446266162203303177}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{18} + \frac{69387061058641530842320147786395780343724302585162534926494429599698906200673150727266150}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{17} - \frac{103671710615111908736802218626860494267430279148077897593145249990960332700783410862925831}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{16} - \frac{56722135556216473036259675884641581571471576561177941026786521326574358835724002932000670}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{15} - \frac{5425308033372228701169275441527801648722971552512741093414525520521252061326187060241471}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{14} + \frac{24324217060928721124428191040451664702652991792968428387439670123279576647597951893569302}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{13} - \frac{78263950362841459729787196317648739463687296711287433777533251537705787307284026060914137}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{12} - \frac{80866449050462395696278813655264069219166262272323700547531714889932988237699413827508352}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{11} + \frac{96830696799699989912495179758971057121613384230549645140161367123130209634596670599403031}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{10} - \frac{96170816836349740334898488164139728725500946751497123320221150870893378036488715588158513}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{9} + \frac{28937400378703690031847611825841140317301012894583928113020700185300856182846705111392006}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{8} - \frac{28025380631516557522821197498339086063461169733277964222076233955895291409777006841775979}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{7} - \frac{48368725984565012031422000127768956888900545064172118138802509506935538637125025749643952}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{6} + \frac{26824145672880604505034366054112066976704635635395981781423912420264563507393337700340295}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{5} - \frac{11778678186016523075402969732784372000921788393273931283744042860607009513615901262360968}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{4} + \frac{8056048615987693386949172705026503780688965462125282483234929247137107852674317913043270}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{3} + \frac{102816985514045985489094340150441040983980836211144887110488205764799464190589919103979972}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a^{2} + \frac{5598773609295649000929378648953696074453590777708777546631006642613966328100181881065126}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201} a - \frac{70163926087019676416030704269984642082137722638972853022792515525070316615543556083895143}{224058951994527002808515139261650472024042742454560016087032251019777696027502780378546201}$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{28}$, which has order $3584$ (assuming GRH)
Unit group
| Rank: | $13$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | 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: | \( 935057598520.6207 \) (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
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 | ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ | R | ${\href{/LocalNumberField/11.7.0.1}{7} }^{4}$ | ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/17.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ | ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{14}$ | ${\href{/LocalNumberField/41.14.0.1}{14} }^{2}$ | R | ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ | ${\href{/LocalNumberField/59.14.0.1}{14} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $7$ | 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $43$ | 43.14.13.11 | $x^{14} + 205667667$ | $14$ | $1$ | $13$ | $C_{14}$ | $[\ ]_{14}$ |
| 43.14.13.11 | $x^{14} + 205667667$ | $14$ | $1$ | $13$ | $C_{14}$ | $[\ ]_{14}$ |