Normalized defining polynomial
\( x^{12} + 93x^{10} + 4164x^{8} + 115271x^{6} + 1988841x^{4} + 19219578x^{2} + 78446449 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(240148397880610960674816\)
\(\medspace = 2^{12}\cdot 3^{18}\cdot 73^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(88.79\) | 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: | $2\cdot 3^{3/2}73^{1/2}\approx 88.79189152169245$ | ||
Ramified primes: |
\(2\), \(3\), \(73\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(2628=2^{2}\cdot 3^{2}\cdot 73\) | ||
Dirichlet character group: | $\lbrace$$\chi_{2628}(1,·)$, $\chi_{2628}(1459,·)$, $\chi_{2628}(1313,·)$, $\chi_{2628}(1607,·)$, $\chi_{2628}(583,·)$, $\chi_{2628}(2189,·)$, $\chi_{2628}(877,·)$, $\chi_{2628}(2483,·)$, $\chi_{2628}(437,·)$, $\chi_{2628}(1753,·)$, $\chi_{2628}(731,·)$, $\chi_{2628}(2335,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-219}) \), \(\Q(\sqrt{-73}) \), 6.0.7657021611.2$^{3}$, 6.0.163349794368.4$^{3}$, 12.0.240148397880610960674816.1$^{24}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{37}a^{8}-\frac{1}{37}a^{6}+\frac{13}{37}a^{4}+\frac{5}{37}a^{2}+\frac{10}{37}$, $\frac{1}{37}a^{9}-\frac{1}{37}a^{7}+\frac{13}{37}a^{5}+\frac{5}{37}a^{3}+\frac{10}{37}a$, $\frac{1}{6273924499}a^{10}-\frac{52762380}{6273924499}a^{8}-\frac{2132892469}{6273924499}a^{6}+\frac{2597138409}{6273924499}a^{4}+\frac{1642165230}{6273924499}a^{2}-\frac{473107472}{6273924499}$, $\frac{1}{55568149287643}a^{11}+\frac{96599588010}{55568149287643}a^{9}+\frac{21652910995174}{55568149287643}a^{7}+\frac{26549267348948}{55568149287643}a^{5}+\frac{8986071613325}{55568149287643}a^{3}-\frac{13840411421212}{55568149287643}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{364}$, which has order $11648$ (assuming GRH)
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{55524}{6273924499}a^{10}+\frac{5222859}{6273924499}a^{8}+\frac{216525422}{6273924499}a^{6}+\frac{5497557097}{6273924499}a^{4}+\frac{80502849770}{6273924499}a^{2}+\frac{13476358145}{169565527}$, $\frac{108898}{6273924499}a^{10}+\frac{10225155}{6273924499}a^{8}+\frac{421747006}{6273924499}a^{6}+\frac{10601207900}{6273924499}a^{4}+\frac{153752540573}{6273924499}a^{2}+\frac{930229587892}{6273924499}$, $\frac{547784701}{55568149287643}a^{11}+\frac{41568320130}{55568149287643}a^{9}+\frac{1578305860528}{55568149287643}a^{7}+\frac{36619564267498}{55568149287643}a^{5}+\frac{478053801219517}{55568149287643}a^{3}+\frac{26\!\cdots\!71}{55568149287643}a+1$, $\frac{1503151902}{55568149287643}a^{11}-\frac{108898}{6273924499}a^{10}+\frac{111699714870}{55568149287643}a^{9}-\frac{10225155}{6273924499}a^{8}+\frac{4160845146261}{55568149287643}a^{7}-\frac{421747006}{6273924499}a^{6}+\frac{94875579170265}{55568149287643}a^{5}-\frac{10601207900}{6273924499}a^{4}+\frac{11\!\cdots\!94}{55568149287643}a^{3}-\frac{153752540573}{6273924499}a^{2}+\frac{61\!\cdots\!11}{55568149287643}a-\frac{930229587892}{6273924499}$, $\frac{547784701}{55568149287643}a^{11}+\frac{164422}{6273924499}a^{10}+\frac{41568320130}{55568149287643}a^{9}+\frac{15448014}{6273924499}a^{8}+\frac{1578305860528}{55568149287643}a^{7}+\frac{638272428}{6273924499}a^{6}+\frac{36619564267498}{55568149287643}a^{5}+\frac{16098764997}{6273924499}a^{4}+\frac{478053801219517}{55568149287643}a^{3}+\frac{234255390343}{6273924499}a^{2}+\frac{26\!\cdots\!71}{55568149287643}a+\frac{1428854839257}{6273924499}$
| 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: | \( 325.67540279491664 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 325.67540279491664 \cdot 11648}{2\cdot\sqrt{240148397880610960674816}}\cr\approx \mathstrut & 0.238147314548371 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times C_6$ (as 12T2):
An abelian group of order 12 |
The 12 conjugacy class representatives for $C_6\times C_2$ |
Character table for $C_6\times C_2$ |
Intermediate fields
\(\Q(\sqrt{-73}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-219}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{3}, \sqrt{-73})\), 6.0.163349794368.4, \(\Q(\zeta_{36})^+\), 6.0.7657021611.2 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.1.0.1}{1} }^{12}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ |
\(3\)
| 3.12.18.82 | $x^{12} + 24 x^{11} + 252 x^{10} + 1558 x^{9} + 6450 x^{8} + 19068 x^{7} + 41627 x^{6} + 68094 x^{5} + 83298 x^{4} + 74306 x^{3} + 45618 x^{2} + 17400 x + 3277$ | $6$ | $2$ | $18$ | $C_6\times C_2$ | $[2]_{2}^{2}$ |
\(73\)
| 73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |