Normalized defining polynomial
\( x^{12} - 3 x^{11} - 2 x^{10} + x^{9} + 36 x^{8} - 23 x^{7} - 116 x^{6} - 10 x^{5} + 319 x^{4} + 209 x^{3} - 473 x^{2} - 247 x + 169 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(3429038075724121\) \(\medspace = 7^{8}\cdot 29^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(19.71\) | 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))
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Galois root discriminant: | $7^{2/3}29^{1/2}\approx 19.705964328162043$ | ||
Ramified primes: | \(7\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3251854344479}a^{11}+\frac{1339303801404}{3251854344479}a^{10}-\frac{650708486781}{3251854344479}a^{9}+\frac{444497570622}{3251854344479}a^{8}+\frac{737894480411}{3251854344479}a^{7}-\frac{1601948339807}{3251854344479}a^{6}-\frac{377725152648}{3251854344479}a^{5}+\frac{199110752051}{3251854344479}a^{4}-\frac{10086624021}{3251854344479}a^{3}+\frac{774023811861}{3251854344479}a^{2}+\frac{1460316099341}{3251854344479}a+\frac{5171728660}{250142641883}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $\frac{56106389219}{3251854344479}a^{11}-\frac{174958974662}{3251854344479}a^{10}-\frac{340210787}{3251854344479}a^{9}-\frac{166765475269}{3251854344479}a^{8}+\frac{1891909408047}{3251854344479}a^{7}-\frac{1910844730960}{3251854344479}a^{6}-\frac{3434369672867}{3251854344479}a^{5}-\frac{1268405889253}{3251854344479}a^{4}+\frac{11789137748016}{3251854344479}a^{3}+\frac{4339535272797}{3251854344479}a^{2}-\frac{13034481263451}{3251854344479}a+\frac{161219520514}{250142641883}$, $\frac{58803426484}{3251854344479}a^{11}-\frac{177405663322}{3251854344479}a^{10}-\frac{23785815580}{3251854344479}a^{9}-\frac{179420819568}{3251854344479}a^{8}+\frac{2018636976053}{3251854344479}a^{7}-\frac{1734339135538}{3251854344479}a^{6}-\frac{3972636236416}{3251854344479}a^{5}-\frac{2370330546346}{3251854344479}a^{4}+\frac{13045666786482}{3251854344479}a^{3}+\frac{7373138608716}{3251854344479}a^{2}-\frac{12581484433854}{3251854344479}a-\frac{102922591481}{250142641883}$, $\frac{431095498}{250142641883}a^{11}-\frac{177614125}{250142641883}a^{10}-\frac{4145848903}{250142641883}a^{9}-\frac{1710346819}{250142641883}a^{8}+\frac{15686413085}{250142641883}a^{7}+\frac{20579678181}{250142641883}a^{6}-\frac{54703212980}{250142641883}a^{5}-\frac{101172304383}{250142641883}a^{4}+\frac{127606949367}{250142641883}a^{3}+\frac{268309536382}{250142641883}a^{2}+\frac{31207780279}{250142641883}a+\frac{68451774258}{250142641883}$, $\frac{29199783557}{3251854344479}a^{11}-\frac{70702201935}{3251854344479}a^{10}-\frac{65016107863}{3251854344479}a^{9}-\frac{137140237082}{3251854344479}a^{8}+\frac{1071775580940}{3251854344479}a^{7}-\frac{187240917460}{3251854344479}a^{6}-\frac{2502127258492}{3251854344479}a^{5}-\frac{3259607601964}{3251854344479}a^{4}+\frac{6121005968413}{3251854344479}a^{3}+\frac{11359356968904}{3251854344479}a^{2}-\frac{6552911658000}{3251854344479}a-\frac{697586998188}{250142641883}$, $\frac{205014965034}{3251854344479}a^{11}-\frac{535665092951}{3251854344479}a^{10}-\frac{237240887148}{3251854344479}a^{9}-\frac{861903657671}{3251854344479}a^{8}+\frac{6534104707766}{3251854344479}a^{7}-\frac{3671876923012}{3251854344479}a^{6}-\frac{13237724533635}{3251854344479}a^{5}-\frac{13742891253922}{3251854344479}a^{4}+\frac{35386394199856}{3251854344479}a^{3}+\frac{31137407686446}{3251854344479}a^{2}-\frac{20400363329846}{3251854344479}a+\frac{17331678044}{250142641883}$, $\frac{58563646917}{3251854344479}a^{11}-\frac{258755355119}{3251854344479}a^{10}+\frac{233374829215}{3251854344479}a^{9}-\frac{216916466069}{3251854344479}a^{8}+\frac{2362712089399}{3251854344479}a^{7}-\frac{4484934294294}{3251854344479}a^{6}-\frac{1198584091202}{3251854344479}a^{5}+\frac{2365649789894}{3251854344479}a^{4}+\frac{15969062053618}{3251854344479}a^{3}-\frac{7441563958365}{3251854344479}a^{2}-\frac{22014745962241}{3251854344479}a+\frac{681767401841}{250142641883}$, $\frac{18164859367}{3251854344479}a^{11}-\frac{57191615366}{3251854344479}a^{10}-\frac{33883030074}{3251854344479}a^{9}+\frac{41610464160}{3251854344479}a^{8}+\frac{666590281511}{3251854344479}a^{7}-\frac{544519333447}{3251854344479}a^{6}-\frac{2283629281994}{3251854344479}a^{5}+\frac{356617969879}{3251854344479}a^{4}+\frac{6896514795166}{3251854344479}a^{3}+\frac{2539926569237}{3251854344479}a^{2}-\frac{14877436160989}{3251854344479}a-\frac{880263521708}{250142641883}$ | 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: | \( 1824.19500337 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 1824.19500337 \cdot 1}{2\cdot\sqrt{3429038075724121}}\cr\approx \mathstrut & 0.388413725663 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 12T7):
A solvable group of order 24 |
The 8 conjugacy class representatives for $A_4 \times C_2$ |
Character table for $A_4 \times C_2$ |
Intermediate fields
\(\Q(\sqrt{29}) \), \(\Q(\zeta_{7})^+\), 6.2.69629.1, 6.6.58557989.1, 6.2.2019241.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 6 sibling: | 6.2.69629.1 |
Degree 8 sibling: | 8.0.1698181681.2 |
Degree 12 sibling: | 12.0.4077334216081.1 |
Minimal sibling: | 6.2.69629.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
\(29\) | 29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |