Normalized defining polynomial
\( x^{12} - 2 x^{11} + 2 x^{10} - 14 x^{9} - 60 x^{8} + 36 x^{7} + 146 x^{6} + 268 x^{5} + 936 x^{4} + 1316 x^{3} + 1152 x^{2} + 1056 x + 484 \)
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: | \(289254654976000000\) \(\medspace = 2^{16}\cdot 5^{6}\cdot 7^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(28.52\) | 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: | $2^{4/3}5^{1/2}7^{5/6}\approx 28.517187846962024$ | ||
Ramified primes: | \(2\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{246}a^{10}+\frac{5}{82}a^{9}+\frac{17}{82}a^{8}-\frac{23}{246}a^{7}-\frac{5}{123}a^{6}-\frac{35}{123}a^{5}+\frac{16}{123}a^{4}-\frac{10}{123}a^{3}-\frac{46}{123}a^{2}-\frac{32}{123}a+\frac{37}{123}$, $\frac{1}{29019206238}a^{11}-\frac{772973}{707785518}a^{10}+\frac{383066292}{4836534373}a^{9}+\frac{3500025368}{14509603119}a^{8}-\frac{1660023802}{14509603119}a^{7}-\frac{1387596033}{9673068746}a^{6}+\frac{985753764}{4836534373}a^{5}+\frac{6747052369}{14509603119}a^{4}-\frac{512346393}{4836534373}a^{3}-\frac{3698711227}{14509603119}a^{2}+\frac{1539225220}{4836534373}a-\frac{448759106}{1319054829}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{6}$, which has order $6$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{10782023}{235928506} a^{11} - \frac{16125900}{117964253} a^{10} + \frac{53426107}{235928506} a^{9} - \frac{101677030}{117964253} a^{8} - \frac{222652114}{117964253} a^{7} + \frac{415754024}{117964253} a^{6} + \frac{380392390}{117964253} a^{5} + \frac{1051946014}{117964253} a^{4} + \frac{3983952406}{117964253} a^{3} + \frac{3102320718}{117964253} a^{2} + \frac{2966020988}{117964253} a + \frac{237351277}{10724023} \) (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: | $\frac{262874185}{9673068746}a^{11}-\frac{2094867347}{29019206238}a^{10}+\frac{492483235}{4836534373}a^{9}-\frac{2160566125}{4836534373}a^{8}-\frac{39056968607}{29019206238}a^{7}+\frac{27457463152}{14509603119}a^{6}+\frac{38533527514}{14509603119}a^{5}+\frac{81835091053}{14509603119}a^{4}+\frac{318267823217}{14509603119}a^{3}+\frac{301460933069}{14509603119}a^{2}+\frac{256192566247}{14509603119}a+\frac{22669828907}{1319054829}$, $\frac{1731886447}{29019206238}a^{11}-\frac{5279801845}{29019206238}a^{10}+\frac{1512013369}{4836534373}a^{9}-\frac{16813131667}{14509603119}a^{8}-\frac{69217927397}{29019206238}a^{7}+\frac{22523784148}{4836534373}a^{6}+\frac{17169871518}{4836534373}a^{5}+\frac{173091388810}{14509603119}a^{4}+\frac{214312132781}{4836534373}a^{3}+\frac{481217776601}{14509603119}a^{2}+\frac{172632266711}{4836534373}a+\frac{42128465905}{1319054829}$, $\frac{7177145}{879369886}a^{11}-\frac{11074926}{439684943}a^{10}+\frac{17849810}{439684943}a^{9}-\frac{64488424}{439684943}a^{8}-\frac{146020517}{439684943}a^{7}+\frac{289587598}{439684943}a^{6}+\frac{290480850}{439684943}a^{5}+\frac{311984746}{439684943}a^{4}+\frac{2759931068}{439684943}a^{3}+\frac{2242681759}{439684943}a^{2}+\frac{2202407262}{439684943}a+\frac{2429805599}{439684943}$, $\frac{54728438}{1319054829}a^{11}-\frac{150396872}{1319054829}a^{10}+\frac{145422357}{879369886}a^{9}-\frac{1825854227}{2638109658}a^{8}-\frac{2648272630}{1319054829}a^{7}+\frac{33244078}{10724023}a^{6}+\frac{1638679288}{439684943}a^{5}+\frac{10742045998}{1319054829}a^{4}+\frac{14584880891}{439684943}a^{3}+\frac{37722655517}{1319054829}a^{2}+\frac{10988568609}{439684943}a+\frac{32338435295}{1319054829}$, $\frac{229625801}{14509603119}a^{11}-\frac{226068915}{4836534373}a^{10}+\frac{366794995}{4836534373}a^{9}-\frac{8617746983}{29019206238}a^{8}-\frac{18909010837}{29019206238}a^{7}+\frac{17308489918}{14509603119}a^{6}+\frac{18109201258}{14509603119}a^{5}+\frac{1125280342}{353892759}a^{4}+\frac{159518035202}{14509603119}a^{3}+\frac{126070780954}{14509603119}a^{2}+\frac{129773268796}{14509603119}a+\frac{3206618023}{439684943}$ | 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: | \( 1815.0374540202486 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1815.0374540202486 \cdot 6}{4\cdot\sqrt{289254654976000000}}\cr\approx \mathstrut & 0.311469755618998 \end{aligned}\]
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
\(\Q(\sqrt{35}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{35})\), 6.0.33614000.1 |
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: | 18.6.155568095557812224000000000.1, 18.0.177792109208928256000000.1 |
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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{3}$ | R | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{6}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 | $x^{12} + 4 x^{11} + 8 x^{10} + 10 x^{9} + 8 x^{8} + 4 x^{7} + 2 x^{6} - 4 x^{5} - 4 x^{4} + 4 x^{3} + 4$ | $6$ | $2$ | $16$ | $C_6\times S_3$ | $[2]_{3}^{6}$ |
\(5\) | 5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(7\) | 7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |
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) | $1$ | $-1$ |
* | 1.35.2t1.a.a | $1$ | $ 5 \cdot 7 $ | \(\Q(\sqrt{-35}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.140.2t1.a.a | $1$ | $ 2^{2} \cdot 5 \cdot 7 $ | \(\Q(\sqrt{35}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.28.6t1.a.a | $1$ | $ 2^{2} \cdot 7 $ | 6.0.153664.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.140.6t1.a.a | $1$ | $ 2^{2} \cdot 5 \cdot 7 $ | 6.6.134456000.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
1.35.6t1.a.a | $1$ | $ 5 \cdot 7 $ | 6.0.2100875.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.28.6t1.a.b | $1$ | $ 2^{2} \cdot 7 $ | 6.0.153664.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.7.3t1.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) | $0$ | $1$ | |
1.35.6t1.a.b | $1$ | $ 5 \cdot 7 $ | 6.0.2100875.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.140.6t1.a.b | $1$ | $ 2^{2} \cdot 5 \cdot 7 $ | 6.6.134456000.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
2.140.3t2.a.a | $2$ | $ 2^{2} \cdot 5 \cdot 7 $ | 3.1.140.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.560.6t3.b.a | $2$ | $ 2^{4} \cdot 5 \cdot 7 $ | 6.0.313600.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
* | 2.3920.12t18.b.a | $2$ | $ 2^{4} \cdot 5 \cdot 7^{2}$ | 12.0.289254654976000000.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
* | 2.980.6t5.a.a | $2$ | $ 2^{2} \cdot 5 \cdot 7^{2}$ | 6.0.33614000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
* | 2.3920.12t18.b.b | $2$ | $ 2^{4} \cdot 5 \cdot 7^{2}$ | 12.0.289254654976000000.2 | $C_6\times S_3$ (as 12T18) | $0$ | $0$ |
* | 2.980.6t5.a.b | $2$ | $ 2^{2} \cdot 5 \cdot 7^{2}$ | 6.0.33614000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |