Normalized defining polynomial
\( x^{36} - 38x^{30} + 1315x^{24} - 4900x^{18} + 16603x^{12} - 129x^{6} + 1 \)
Invariants
Degree: | $36$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 18]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(765562336274603149526276140236591524202950795854016937984\) \(\medspace = 2^{36}\cdot 3^{54}\cdot 7^{24}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(38.03\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $2\cdot 3^{3/2}7^{2/3}\approx 38.02862046112015$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $36$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(252=2^{2}\cdot 3^{2}\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{252}(1,·)$, $\chi_{252}(107,·)$, $\chi_{252}(65,·)$, $\chi_{252}(137,·)$, $\chi_{252}(11,·)$, $\chi_{252}(149,·)$, $\chi_{252}(23,·)$, $\chi_{252}(25,·)$, $\chi_{252}(155,·)$, $\chi_{252}(29,·)$, $\chi_{252}(163,·)$, $\chi_{252}(37,·)$, $\chi_{252}(169,·)$, $\chi_{252}(43,·)$, $\chi_{252}(179,·)$, $\chi_{252}(53,·)$, $\chi_{252}(151,·)$, $\chi_{252}(191,·)$, $\chi_{252}(193,·)$, $\chi_{252}(67,·)$, $\chi_{252}(197,·)$, $\chi_{252}(71,·)$, $\chi_{252}(205,·)$, $\chi_{252}(79,·)$, $\chi_{252}(211,·)$, $\chi_{252}(85,·)$, $\chi_{252}(221,·)$, $\chi_{252}(95,·)$, $\chi_{252}(233,·)$, $\chi_{252}(235,·)$, $\chi_{252}(109,·)$, $\chi_{252}(239,·)$, $\chi_{252}(113,·)$, $\chi_{252}(247,·)$, $\chi_{252}(121,·)$, $\chi_{252}(127,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{131072}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{559}a^{24}-\frac{12}{43}a^{18}+\frac{209}{559}a^{12}-\frac{5}{43}a^{6}+\frac{274}{559}$, $\frac{1}{559}a^{25}-\frac{12}{43}a^{19}+\frac{209}{559}a^{13}-\frac{5}{43}a^{7}+\frac{274}{559}a$, $\frac{1}{559}a^{26}-\frac{12}{43}a^{20}+\frac{209}{559}a^{14}-\frac{5}{43}a^{8}+\frac{274}{559}a^{2}$, $\frac{1}{559}a^{27}-\frac{12}{43}a^{21}+\frac{209}{559}a^{15}-\frac{5}{43}a^{9}+\frac{274}{559}a^{3}$, $\frac{1}{559}a^{28}-\frac{12}{43}a^{22}+\frac{209}{559}a^{16}-\frac{5}{43}a^{10}+\frac{274}{559}a^{4}$, $\frac{1}{559}a^{29}-\frac{12}{43}a^{23}+\frac{209}{559}a^{17}-\frac{5}{43}a^{11}+\frac{274}{559}a^{5}$, $\frac{1}{12201876037}a^{30}-\frac{7467523}{12201876037}a^{24}-\frac{114384399}{12201876037}a^{18}+\frac{5503538510}{12201876037}a^{12}+\frac{1127590751}{12201876037}a^{6}+\frac{5582733509}{12201876037}$, $\frac{1}{12201876037}a^{31}-\frac{7467523}{12201876037}a^{25}-\frac{114384399}{12201876037}a^{19}+\frac{5503538510}{12201876037}a^{13}+\frac{1127590751}{12201876037}a^{7}+\frac{5582733509}{12201876037}a$, $\frac{1}{12201876037}a^{32}-\frac{7467523}{12201876037}a^{26}-\frac{114384399}{12201876037}a^{20}+\frac{5503538510}{12201876037}a^{14}+\frac{1127590751}{12201876037}a^{8}+\frac{5582733509}{12201876037}a^{2}$, $\frac{1}{12201876037}a^{33}-\frac{7467523}{12201876037}a^{27}-\frac{114384399}{12201876037}a^{21}+\frac{5503538510}{12201876037}a^{15}+\frac{1127590751}{12201876037}a^{9}+\frac{5582733509}{12201876037}a^{3}$, $\frac{1}{12201876037}a^{34}-\frac{7467523}{12201876037}a^{28}-\frac{114384399}{12201876037}a^{22}+\frac{5503538510}{12201876037}a^{16}+\frac{1127590751}{12201876037}a^{10}+\frac{5582733509}{12201876037}a^{4}$, $\frac{1}{12201876037}a^{35}-\frac{7467523}{12201876037}a^{29}-\frac{114384399}{12201876037}a^{23}+\frac{5503538510}{12201876037}a^{17}+\frac{1127590751}{12201876037}a^{11}+\frac{5582733509}{12201876037}a^{5}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{14}$, which has order $28$ (assuming GRH)
Unit group
Rank: | $17$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{5430336179}{12201876037} a^{35} + \frac{206353143668}{12201876037} a^{29} - \frac{7140904840090}{12201876037} a^{23} + \frac{26609083833276}{12201876037} a^{17} - \frac{90160032745258}{12201876037} a^{11} + \frac{16291037658}{283764559} a^{5} \) (order $36$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{7075524}{12201876037}a^{35}-\frac{244850370}{12201876037}a^{29}+\frac{8391543538}{12201876037}a^{23}-\frac{3091445394}{12201876037}a^{17}+\frac{558594}{283764559}a^{11}+\frac{387866503351}{12201876037}a^{5}$, $\frac{198436}{938605849}a^{34}-\frac{6866930}{938605849}a^{28}+\frac{235334231}{938605849}a^{22}-\frac{86700866}{938605849}a^{16}+\frac{15666}{21828043}a^{10}+\frac{10648303918}{938605849}a^{4}$, $\frac{2416738802}{12201876037}a^{34}-\frac{91835779862}{12201876037}a^{28}+\frac{3178001329435}{12201876037}a^{22}-\frac{11841670194505}{12201876037}a^{16}+\frac{40124985606547}{12201876037}a^{10}-\frac{7250193147}{283764559}a^{4}+1$, $\frac{5453950}{12201876037}a^{34}+\frac{11154294}{283764559}a^{32}-\frac{188735375}{12201876037}a^{28}-\frac{18222348943}{12201876037}a^{26}+\frac{6468625301}{12201876037}a^{22}+\frac{630578266506}{12201876037}a^{20}-\frac{2382945575}{12201876037}a^{16}-\frac{2345320446930}{12201876037}a^{14}+\frac{430575}{283764559}a^{10}+\frac{7947244975065}{12201876037}a^{8}+\frac{311044791177}{12201876037}a^{4}-\frac{141287724}{12201876037}a^{2}$, $\frac{5453950}{12201876037}a^{34}-\frac{4969104}{283764559}a^{31}-\frac{188735375}{12201876037}a^{28}+\frac{8117929470}{12201876037}a^{25}+\frac{6468625301}{12201876037}a^{22}-\frac{280915043696}{12201876037}a^{19}-\frac{2382945575}{12201876037}a^{16}+\frac{1044812088880}{12201876037}a^{13}+\frac{430575}{283764559}a^{10}-\frac{3536864877885}{12201876037}a^{7}+\frac{311044791177}{12201876037}a^{4}+\frac{62941984}{12201876037}a$, $\frac{5453950}{12201876037}a^{34}+\frac{82733471}{938605849}a^{33}-\frac{188735375}{12201876037}a^{28}-\frac{3143905984}{938605849}a^{27}+\frac{6468625301}{12201876037}a^{22}+\frac{108795693920}{938605849}a^{21}-\frac{2382945575}{12201876037}a^{16}-\frac{405434252289}{938605849}a^{15}+\frac{430575}{283764559}a^{10}+\frac{1373638711904}{938605849}a^{9}+\frac{311044791177}{12201876037}a^{4}-\frac{248203104}{21828043}a^{3}$, $\frac{12529474}{12201876037}a^{35}-\frac{433585745}{12201876037}a^{29}+\frac{14860168839}{12201876037}a^{23}-\frac{5474390969}{12201876037}a^{17}+\frac{989169}{283764559}a^{11}+\frac{698911294528}{12201876037}a^{5}-1$, $\frac{9949806}{12201876037}a^{34}+\frac{958094}{12201876037}a^{32}+\frac{294614}{12201876037}a^{30}-\frac{344315655}{12201876037}a^{28}-\frac{33155095}{12201876037}a^{26}-\frac{10195195}{12201876037}a^{24}+\frac{11800823836}{12201876037}a^{22}+\frac{1136426766}{12201876037}a^{20}+\frac{349935295}{12201876037}a^{18}-\frac{4347279711}{12201876037}a^{16}-\frac{418611439}{12201876037}a^{14}-\frac{128723059}{12201876037}a^{12}+\frac{785511}{283764559}a^{10}+\frac{75639}{283764559}a^{8}+\frac{23259}{283764559}a^{6}+\frac{560483343594}{12201876037}a^{4}+\frac{49404362723}{12201876037}a^{2}+\frac{21987013272}{12201876037}$, $\frac{12529474}{12201876037}a^{35}-\frac{198436}{938605849}a^{33}-\frac{169635}{21828043}a^{31}-\frac{433585745}{12201876037}a^{29}+\frac{6866930}{938605849}a^{27}+\frac{6444686}{21828043}a^{25}+\frac{14860168839}{12201876037}a^{23}-\frac{235334231}{938605849}a^{21}-\frac{223020055}{21828043}a^{19}-\frac{5474390969}{12201876037}a^{17}+\frac{86700866}{938605849}a^{15}+\frac{829482275}{21828043}a^{13}+\frac{989169}{283764559}a^{11}-\frac{15666}{21828043}a^{9}-\frac{2815818991}{21828043}a^{7}+\frac{698911294528}{12201876037}a^{5}-\frac{10648303918}{938605849}a^{3}+\frac{21878013}{21828043}a$, $\frac{22479280}{12201876037}a^{35}-\frac{82733471}{938605849}a^{33}+\frac{213007992}{12201876037}a^{31}-\frac{777901400}{12201876037}a^{29}+\frac{3143905984}{938605849}a^{27}-\frac{8094969570}{12201876037}a^{25}+\frac{26660992675}{12201876037}a^{23}-\frac{108795693920}{938605849}a^{21}+\frac{280128552225}{12201876037}a^{19}-\frac{9821670680}{12201876037}a^{17}+\frac{405434252289}{938605849}a^{15}-\frac{1044522200500}{12201876037}a^{13}+\frac{1774680}{283764559}a^{11}-\frac{1373638711904}{938605849}a^{9}+\frac{3536862625545}{12201876037}a^{7}+\frac{1259394638122}{12201876037}a^{5}+\frac{248203104}{21828043}a^{3}-\frac{639076545}{283764559}a$, $\frac{12529474}{12201876037}a^{35}-\frac{82733471}{938605849}a^{33}-\frac{294614}{12201876037}a^{31}-\frac{433585745}{12201876037}a^{29}+\frac{3143905984}{938605849}a^{27}+\frac{10195195}{12201876037}a^{25}+\frac{14860168839}{12201876037}a^{23}-\frac{108795693920}{938605849}a^{21}-\frac{349935295}{12201876037}a^{19}-\frac{5474390969}{12201876037}a^{17}+\frac{405434252289}{938605849}a^{15}+\frac{128723059}{12201876037}a^{13}+\frac{989169}{283764559}a^{11}-\frac{1373638711904}{938605849}a^{9}-\frac{23259}{283764559}a^{7}+\frac{698911294528}{12201876037}a^{5}+\frac{248203104}{21828043}a^{3}-\frac{34188889309}{12201876037}a$, $\frac{12529474}{12201876037}a^{35}+\frac{4354801056}{12201876037}a^{34}-\frac{579}{6553}a^{33}-\frac{433585745}{12201876037}a^{29}-\frac{165482365876}{12201876037}a^{28}+\frac{945894}{281779}a^{27}+\frac{14860168839}{12201876037}a^{23}+\frac{5726560819130}{12201876037}a^{22}-\frac{32732221}{281779}a^{21}-\frac{5474390969}{12201876037}a^{17}-\frac{21338438553519}{12201876037}a^{16}+\frac{121741505}{281779}a^{15}+\frac{989169}{283764559}a^{11}+\frac{72302729490506}{12201876037}a^{10}-\frac{412380482}{281779}a^{9}+\frac{698911294528}{12201876037}a^{5}-\frac{13064397306}{283764559}a^{4}+\frac{7334}{281779}a^{3}$, $\frac{11154294}{283764559}a^{32}-\frac{8949039}{283764559}a^{31}+\frac{169635}{21828043}a^{30}-\frac{18222348943}{12201876037}a^{26}+\frac{14619769469}{12201876037}a^{25}-\frac{6444686}{21828043}a^{24}+\frac{630578266506}{12201876037}a^{20}-\frac{505910055761}{12201876037}a^{19}+\frac{223020055}{21828043}a^{18}-\frac{2345320446930}{12201876037}a^{14}+\frac{1881639855205}{12201876037}a^{13}-\frac{829482275}{21828043}a^{12}+\frac{7947244975065}{12201876037}a^{8}-\frac{6373202159096}{12201876037}a^{7}+\frac{2815818991}{21828043}a^{6}-\frac{141287724}{12201876037}a^{2}+\frac{113354494}{12201876037}a-\frac{49970}{21828043}$, $\frac{12529474}{12201876037}a^{35}-\frac{198436}{938605849}a^{34}+\frac{198436}{938605849}a^{33}-\frac{433585745}{12201876037}a^{29}+\frac{6866930}{938605849}a^{28}-\frac{6866930}{938605849}a^{27}+\frac{14860168839}{12201876037}a^{23}-\frac{235334231}{938605849}a^{22}+\frac{235334231}{938605849}a^{21}-\frac{5474390969}{12201876037}a^{17}+\frac{86700866}{938605849}a^{16}-\frac{86700866}{938605849}a^{15}+\frac{989169}{283764559}a^{11}-\frac{15666}{21828043}a^{10}+\frac{15666}{21828043}a^{9}+\frac{698911294528}{12201876037}a^{5}-\frac{10648303918}{938605849}a^{4}+\frac{10648303918}{938605849}a^{3}$, $\frac{958094}{12201876037}a^{32}-\frac{265668556}{12201876037}a^{31}-\frac{169635}{21828043}a^{30}-\frac{33155095}{12201876037}a^{26}+\frac{10094224278}{12201876037}a^{25}+\frac{6444686}{21828043}a^{24}+\frac{1136426766}{12201876037}a^{20}-\frac{349313287515}{12201876037}a^{19}-\frac{223020055}{21828043}a^{18}-\frac{418611439}{12201876037}a^{14}+\frac{1300379634991}{12201876037}a^{13}+\frac{829482275}{21828043}a^{12}+\frac{75639}{283764559}a^{8}-\frac{4410379097043}{12201876037}a^{7}-\frac{2815818991}{21828043}a^{6}+\frac{61606238760}{12201876037}a^{2}+\frac{796912443}{283764559}a+\frac{49970}{21828043}$, $\frac{157835898}{283764559}a^{35}+\frac{2416738802}{12201876037}a^{34}+\frac{169635}{21828043}a^{30}-\frac{257851196768}{12201876037}a^{29}-\frac{91835779862}{12201876037}a^{28}-\frac{6444686}{21828043}a^{24}+\frac{8922831597702}{12201876037}a^{23}+\frac{3178001329435}{12201876037}a^{22}+\frac{223020055}{21828043}a^{18}-\frac{33186838973310}{12201876037}a^{17}-\frac{11841670194505}{12201876037}a^{16}-\frac{829482275}{21828043}a^{12}+\frac{112427767388751}{12201876037}a^{11}+\frac{40124985606547}{12201876037}a^{10}+\frac{2815818991}{21828043}a^{6}-\frac{1999254708}{12201876037}a^{5}-\frac{7250193147}{283764559}a^{4}-\frac{49970}{21828043}$, $\frac{12529474}{12201876037}a^{35}-\frac{213007992}{12201876037}a^{31}-\frac{368866}{12201876037}a^{30}-\frac{433585745}{12201876037}a^{29}+\frac{8094969570}{12201876037}a^{25}+\frac{12764705}{12201876037}a^{24}+\frac{14860168839}{12201876037}a^{23}-\frac{280128552225}{12201876037}a^{19}-\frac{436556176}{12201876037}a^{18}-\frac{5474390969}{12201876037}a^{17}+\frac{1044522200500}{12201876037}a^{13}+\frac{161165321}{12201876037}a^{12}+\frac{989169}{283764559}a^{11}-\frac{3536862625545}{12201876037}a^{7}-\frac{29121}{283764559}a^{6}+\frac{698911294528}{12201876037}a^{5}+\frac{639076545}{283764559}a-\frac{5430336179}{12201876037}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 18271878541587.973 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 18271878541587.973 \cdot 28}{36\cdot\sqrt{765562336274603149526276140236591524202950795854016937984}}\cr\approx \mathstrut & 0.119642665807942 \end{aligned}\] (assuming GRH)
Galois group
An abelian group of order 36 |
The 36 conjugacy class representatives for $C_6^2$ |
Character table for $C_6^2$ |
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 | ${\href{/padicField/5.6.0.1}{6} }^{6}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{6}$ | ${\href{/padicField/13.3.0.1}{3} }^{12}$ | ${\href{/padicField/17.6.0.1}{6} }^{6}$ | ${\href{/padicField/19.6.0.1}{6} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{12}$ | ${\href{/padicField/41.6.0.1}{6} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ |
2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
\(3\) | Deg $36$ | $6$ | $6$ | $54$ | |||
\(7\) | 7.18.12.1 | $x^{18} + 3 x^{16} + 57 x^{15} + 15 x^{14} + 90 x^{13} + 424 x^{12} - 921 x^{11} - 3090 x^{10} - 6496 x^{9} - 10560 x^{8} + 6912 x^{7} + 28033 x^{6} + 33237 x^{5} + 188463 x^{4} - 139476 x^{3} + 351552 x^{2} - 514905 x + 582014$ | $3$ | $6$ | $12$ | $C_6 \times C_3$ | $[\ ]_{3}^{6}$ |
7.18.12.1 | $x^{18} + 3 x^{16} + 57 x^{15} + 15 x^{14} + 90 x^{13} + 424 x^{12} - 921 x^{11} - 3090 x^{10} - 6496 x^{9} - 10560 x^{8} + 6912 x^{7} + 28033 x^{6} + 33237 x^{5} + 188463 x^{4} - 139476 x^{3} + 351552 x^{2} - 514905 x + 582014$ | $3$ | $6$ | $12$ | $C_6 \times C_3$ | $[\ ]_{3}^{6}$ |