Normalized defining polynomial
\( x^{12} + x^{10} + 10x^{8} + 33x^{6} + 50x^{4} + 5x^{2} + 1 \)
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: |
\(7402760456175616\)
\(\medspace = 2^{16}\cdot 7^{4}\cdot 19^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(21.01\) | 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: | $2^{4/3}7^{2/3}19^{1/2}\approx 40.192851765430284$ | ||
Ramified primes: |
\(2\), \(7\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{638}a^{10}-\frac{147}{638}a^{8}+\frac{37}{319}a^{6}-\frac{1}{2}a^{5}-\frac{73}{638}a^{4}-\frac{1}{2}a^{3}-\frac{311}{638}a^{2}-\frac{1}{2}a-\frac{111}{319}$, $\frac{1}{638}a^{11}-\frac{147}{638}a^{9}+\frac{37}{319}a^{7}+\frac{123}{319}a^{5}-\frac{1}{2}a^{4}+\frac{4}{319}a^{3}-\frac{1}{2}a^{2}+\frac{97}{638}a-\frac{1}{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: |
\( \frac{70}{319} a^{11} + \frac{155}{638} a^{9} + \frac{714}{319} a^{7} + \frac{4773}{638} a^{5} + \frac{3750}{319} a^{3} + \frac{1777}{638} a \)
(order $4$)
| 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: |
$\frac{75}{638}a^{11}+\frac{70}{319}a^{9}+\frac{765}{638}a^{7}+\frac{1569}{319}a^{5}+\frac{5385}{638}a^{3}+\frac{1245}{319}a$, $a$, $\frac{677}{638}a^{11}+\frac{149}{638}a^{10}+\frac{647}{638}a^{9}+\frac{54}{319}a^{8}+\frac{3357}{319}a^{7}+\frac{728}{319}a^{6}+\frac{22035}{638}a^{5}+\frac{4435}{638}a^{4}+\frac{32531}{638}a^{3}+\frac{3148}{319}a^{2}+\frac{775}{319}a-\frac{589}{319}$, $\frac{185}{638}a^{11}-\frac{57}{319}a^{10}+\frac{239}{638}a^{9}-\frac{149}{638}a^{8}+\frac{1887}{638}a^{7}-\frac{1099}{638}a^{6}+\frac{3296}{319}a^{5}-\frac{4119}{638}a^{4}+\frac{5525}{319}a^{3}-\frac{3008}{319}a^{2}+\frac{1157}{319}a-\frac{106}{319}$, $\frac{313}{638}a^{11}+\frac{13}{319}a^{10}+\frac{122}{319}a^{9}+\frac{3}{319}a^{8}+\frac{3065}{638}a^{7}+\frac{329}{638}a^{6}+\frac{9689}{638}a^{5}+\frac{327}{319}a^{4}+\frac{6675}{319}a^{3}+\frac{742}{319}a^{2}-\frac{901}{638}a+\frac{927}{638}$
| 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: | \( 583.8867703020712 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 583.8867703020712 \cdot 3}{4\cdot\sqrt{7402760456175616}}\cr\approx \mathstrut & 0.313164309242138 \end{aligned}\]
Galois group
$C_6\times S_3$ (as 12T18):
A solvable group of order 36 |
The 18 conjugacy class representatives for $C_6\times S_3$ |
Character table for $C_6\times S_3$ |
Intermediate fields
\(\Q(\sqrt{19}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{19})\), 6.0.5377456.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 36 |
Degree 18 siblings: | 18.6.74933977343871482567342424064.2, deg 18 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{6}$ | R | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| 2.12.16.3 | $x^{12} + 4 x^{11} + 8 x^{10} + 10 x^{9} + 8 x^{8} + 4 x^{7} + 2 x^{6} - 4 x^{5} - 4 x^{4} + 4 x^{3} + 4$ | $6$ | $2$ | $16$ | $C_6\times S_3$ | $[2]_{3}^{6}$ |
\(7\)
| 7.6.4.1 | $x^{6} + 14 x^{3} - 245$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
7.6.0.1 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(19\)
| 19.12.6.1 | $x^{12} + 114 x^{10} + 34 x^{9} + 5449 x^{8} + 12 x^{7} + 134889 x^{6} - 75118 x^{5} + 1847901 x^{4} - 1865072 x^{3} + 14269503 x^{2} - 12672520 x + 53461691$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.19.2t1.a.a | $1$ | $ 19 $ | \(\Q(\sqrt{-19}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.76.2t1.a.a | $1$ | $ 2^{2} \cdot 19 $ | \(\Q(\sqrt{19}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
1.133.6t1.i.a | $1$ | $ 7 \cdot 19 $ | 6.0.16468459.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.133.6t1.i.b | $1$ | $ 7 \cdot 19 $ | 6.0.16468459.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.532.6t1.h.a | $1$ | $ 2^{2} \cdot 7 \cdot 19 $ | 6.6.1053981376.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
1.28.6t1.a.a | $1$ | $ 2^{2} \cdot 7 $ | 6.0.153664.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.28.6t1.a.b | $1$ | $ 2^{2} \cdot 7 $ | 6.0.153664.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.532.6t1.h.b | $1$ | $ 2^{2} \cdot 7 \cdot 19 $ | 6.6.1053981376.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
2.3724.3t2.b.a | $2$ | $ 2^{2} \cdot 7^{2} \cdot 19 $ | 3.1.3724.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.14896.6t3.k.a | $2$ | $ 2^{4} \cdot 7^{2} \cdot 19 $ | 6.0.221890816.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.2128.12t18.b.a | $2$ | $ 2^{4} \cdot 7 \cdot 19 $ | 12.0.7402760456175616.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
* | 2.532.6t5.c.a | $2$ | $ 2^{2} \cdot 7 \cdot 19 $ | 6.0.5377456.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
* | 2.2128.12t18.b.b | $2$ | $ 2^{4} \cdot 7 \cdot 19 $ | 12.0.7402760456175616.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
* | 2.532.6t5.c.b | $2$ | $ 2^{2} \cdot 7 \cdot 19 $ | 6.0.5377456.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |