Normalized defining polynomial
\( x^{12} - 2 x^{11} + 145 x^{10} - 236 x^{9} + 9374 x^{8} - 12376 x^{7} + 343470 x^{6} - 352786 x^{5} + \cdots + 495287521 \)
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)
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Discriminant: | \(260560131209473946812416\) \(\medspace = 2^{18}\cdot 3^{6}\cdot 7^{10}\cdot 13^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(89.40\) | 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: | $2^{3/2}3^{1/2}7^{5/6}13^{1/2}\approx 89.39755964616802$ | ||
Ramified primes: | \(2\), \(3\), \(7\), \(13\) | 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)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(2184=2^{3}\cdot 3\cdot 7\cdot 13\) | ||
Dirichlet character group: | $\lbrace$$\chi_{2184}(1,·)$, $\chi_{2184}(1091,·)$, $\chi_{2184}(1403,·)$, $\chi_{2184}(1873,·)$, $\chi_{2184}(521,·)$, $\chi_{2184}(1769,·)$, $\chi_{2184}(467,·)$, $\chi_{2184}(1507,·)$, $\chi_{2184}(209,·)$, $\chi_{2184}(883,·)$, $\chi_{2184}(625,·)$, $\chi_{2184}(571,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{-546}) \), 6.0.2700798464.1$^{3}$, 6.0.510450909696.3$^{3}$, 12.0.260560131209473946812416.13$^{24}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{36\!\cdots\!77}a^{11}+\frac{23\!\cdots\!25}{36\!\cdots\!77}a^{10}-\frac{86\!\cdots\!63}{36\!\cdots\!77}a^{9}+\frac{11\!\cdots\!86}{36\!\cdots\!77}a^{8}-\frac{12\!\cdots\!12}{36\!\cdots\!77}a^{7}-\frac{17\!\cdots\!78}{36\!\cdots\!77}a^{6}-\frac{11\!\cdots\!08}{36\!\cdots\!77}a^{5}-\frac{13\!\cdots\!05}{36\!\cdots\!77}a^{4}+\frac{63\!\cdots\!29}{36\!\cdots\!77}a^{3}+\frac{21\!\cdots\!77}{36\!\cdots\!77}a^{2}+\frac{40\!\cdots\!62}{36\!\cdots\!77}a+\frac{65\!\cdots\!15}{36\!\cdots\!77}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{6}\times C_{12}\times C_{12}\times C_{36}$, which has order $31104$ (assuming GRH)
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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: | $\frac{32\!\cdots\!90}{26\!\cdots\!89}a^{11}-\frac{12\!\cdots\!89}{26\!\cdots\!89}a^{10}+\frac{35\!\cdots\!16}{26\!\cdots\!89}a^{9}-\frac{14\!\cdots\!88}{26\!\cdots\!89}a^{8}+\frac{17\!\cdots\!48}{26\!\cdots\!89}a^{7}-\frac{76\!\cdots\!02}{26\!\cdots\!89}a^{6}+\frac{47\!\cdots\!58}{26\!\cdots\!89}a^{5}-\frac{21\!\cdots\!87}{26\!\cdots\!89}a^{4}+\frac{68\!\cdots\!60}{26\!\cdots\!89}a^{3}-\frac{31\!\cdots\!44}{26\!\cdots\!89}a^{2}+\frac{43\!\cdots\!00}{26\!\cdots\!89}a-\frac{19\!\cdots\!47}{26\!\cdots\!89}$, $\frac{71\!\cdots\!50}{26\!\cdots\!89}a^{11}-\frac{14\!\cdots\!27}{26\!\cdots\!89}a^{10}+\frac{10\!\cdots\!90}{26\!\cdots\!89}a^{9}-\frac{16\!\cdots\!44}{26\!\cdots\!89}a^{8}+\frac{64\!\cdots\!40}{26\!\cdots\!89}a^{7}-\frac{89\!\cdots\!61}{26\!\cdots\!89}a^{6}+\frac{21\!\cdots\!22}{26\!\cdots\!89}a^{5}-\frac{25\!\cdots\!61}{26\!\cdots\!89}a^{4}+\frac{38\!\cdots\!50}{26\!\cdots\!89}a^{3}-\frac{39\!\cdots\!03}{26\!\cdots\!89}a^{2}+\frac{29\!\cdots\!66}{26\!\cdots\!89}a-\frac{23\!\cdots\!89}{26\!\cdots\!89}$, $\frac{43\!\cdots\!86}{36\!\cdots\!77}a^{11}-\frac{17\!\cdots\!05}{36\!\cdots\!77}a^{10}+\frac{48\!\cdots\!14}{36\!\cdots\!77}a^{9}-\frac{20\!\cdots\!04}{36\!\cdots\!77}a^{8}+\frac{24\!\cdots\!94}{36\!\cdots\!77}a^{7}-\frac{10\!\cdots\!86}{36\!\cdots\!77}a^{6}+\frac{64\!\cdots\!18}{36\!\cdots\!77}a^{5}-\frac{30\!\cdots\!23}{36\!\cdots\!77}a^{4}+\frac{93\!\cdots\!94}{36\!\cdots\!77}a^{3}-\frac{47\!\cdots\!17}{36\!\cdots\!77}a^{2}+\frac{58\!\cdots\!36}{36\!\cdots\!77}a-\frac{31\!\cdots\!38}{36\!\cdots\!77}$, $\frac{16\!\cdots\!82}{36\!\cdots\!77}a^{11}+\frac{76\!\cdots\!09}{36\!\cdots\!77}a^{10}+\frac{22\!\cdots\!94}{36\!\cdots\!77}a^{9}+\frac{11\!\cdots\!68}{36\!\cdots\!77}a^{8}+\frac{13\!\cdots\!56}{36\!\cdots\!77}a^{7}+\frac{73\!\cdots\!55}{36\!\cdots\!77}a^{6}+\frac{41\!\cdots\!02}{36\!\cdots\!77}a^{5}+\frac{23\!\cdots\!08}{36\!\cdots\!77}a^{4}+\frac{71\!\cdots\!98}{36\!\cdots\!77}a^{3}+\frac{38\!\cdots\!41}{36\!\cdots\!77}a^{2}+\frac{52\!\cdots\!46}{36\!\cdots\!77}a+\frac{27\!\cdots\!97}{36\!\cdots\!77}$, $\frac{52\!\cdots\!36}{36\!\cdots\!77}a^{11}-\frac{95\!\cdots\!80}{36\!\cdots\!77}a^{10}+\frac{77\!\cdots\!34}{36\!\cdots\!77}a^{9}-\frac{11\!\cdots\!59}{36\!\cdots\!77}a^{8}+\frac{51\!\cdots\!24}{36\!\cdots\!77}a^{7}-\frac{58\!\cdots\!01}{36\!\cdots\!77}a^{6}+\frac{19\!\cdots\!20}{36\!\cdots\!77}a^{5}-\frac{16\!\cdots\!08}{36\!\cdots\!77}a^{4}+\frac{48\!\cdots\!34}{36\!\cdots\!77}a^{3}-\frac{25\!\cdots\!17}{36\!\cdots\!77}a^{2}+\frac{52\!\cdots\!34}{36\!\cdots\!77}a+\frac{19\!\cdots\!27}{36\!\cdots\!77}$ (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: | \( 140.7987960054707 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 140.7987960054707 \cdot 31104}{2\cdot\sqrt{260560131209473946812416}}\cr\approx \mathstrut & 0.263943162926681 \end{aligned}\] (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
\(\Q(\sqrt{-546}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-26}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{21}, \sqrt{-26})\), 6.0.510450909696.3, \(\Q(\zeta_{21})^+\), 6.0.2700798464.1 |
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 | R | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.1.0.1}{1} }^{12}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.18.15 | $x^{12} - 12 x^{11} - 56 x^{9} + 2372 x^{8} + 8992 x^{7} + 115648 x^{6} + 164160 x^{5} + 1305648 x^{4} - 282560 x^{3} + 5591296 x^{2} - 9640320 x + 8497856$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ |
\(3\) | 3.6.3.1 | $x^{6} + 18 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
3.6.3.1 | $x^{6} + 18 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(7\) | 7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
\(13\) | 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |