Normalized defining polynomial
\( x^{12} - 3x^{11} + 2x^{10} - 3x^{9} + 8x^{8} - 3x^{7} - 28x^{6} + 24x^{5} + x^{4} + 45x^{3} - 61x^{2} + 3x + 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(27825593350009\) \(\medspace = 7^{8}\cdot 13^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(13.19\) | 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: | $7^{2/3}13^{1/2}\approx 13.193814370085985$ | ||
Ramified primes: | \(7\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2470477409}a^{11}+\frac{490514462}{2470477409}a^{10}-\frac{457346088}{2470477409}a^{9}-\frac{928402176}{2470477409}a^{8}+\frac{556452014}{2470477409}a^{7}+\frac{7887893}{57452963}a^{6}+\frac{127529028}{2470477409}a^{5}+\frac{462448903}{2470477409}a^{4}-\frac{686923154}{2470477409}a^{3}-\frac{152991694}{2470477409}a^{2}+\frac{897962995}{2470477409}a-\frac{236248142}{2470477409}$
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)
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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)
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Fundamental units: | $\frac{284029886}{2470477409}a^{11}-\frac{838357689}{2470477409}a^{10}+\frac{473908449}{2470477409}a^{9}-\frac{771422805}{2470477409}a^{8}+\frac{2340013048}{2470477409}a^{7}-\frac{16998518}{57452963}a^{6}-\frac{7901427513}{2470477409}a^{5}+\frac{6389192247}{2470477409}a^{4}+\frac{1750004301}{2470477409}a^{3}+\frac{14284484761}{2470477409}a^{2}-\frac{17829816795}{2470477409}a+\frac{849176655}{2470477409}$, $\frac{228370815}{2470477409}a^{11}-\frac{744033276}{2470477409}a^{10}+\frac{561019139}{2470477409}a^{9}-\frac{736838163}{2470477409}a^{8}+\frac{2201898037}{2470477409}a^{7}-\frac{21653269}{57452963}a^{6}-\frac{6609153200}{2470477409}a^{5}+\frac{6795022377}{2470477409}a^{4}+\frac{221487386}{2470477409}a^{3}+\frac{12433642340}{2470477409}a^{2}-\frac{17677871651}{2470477409}a-\frac{216861027}{2470477409}$, $\frac{1294397}{57452963}a^{11}-\frac{2193591}{57452963}a^{10}-\frac{2025830}{57452963}a^{9}-\frac{804294}{57452963}a^{8}+\frac{3211977}{57452963}a^{7}+\frac{4654751}{57452963}a^{6}-\frac{30052891}{57452963}a^{5}-\frac{9437910}{57452963}a^{4}+\frac{35546905}{57452963}a^{3}+\frac{43042847}{57452963}a^{2}-\frac{3533608}{57452963}a-\frac{32661389}{57452963}$, $\frac{93920873}{2470477409}a^{11}-\frac{229098580}{2470477409}a^{10}+\frac{31787534}{2470477409}a^{9}-\frac{281568179}{2470477409}a^{8}+\frac{975097613}{2470477409}a^{7}-\frac{6058103}{57452963}a^{6}-\frac{2684227346}{2470477409}a^{5}+\frac{542811006}{2470477409}a^{4}+\frac{1350300016}{2470477409}a^{3}+\frac{8287969744}{2470477409}a^{2}-\frac{8067933126}{2470477409}a-\frac{1835173331}{2470477409}$, $\frac{349430359}{2470477409}a^{11}-\frac{956917818}{2470477409}a^{10}+\frac{546962940}{2470477409}a^{9}-\frac{1098366545}{2470477409}a^{8}+\frac{2528480247}{2470477409}a^{7}-\frac{19643982}{57452963}a^{6}-\frac{9346606537}{2470477409}a^{5}+\frac{5919497077}{2470477409}a^{4}+\frac{623297115}{2470477409}a^{3}+\frac{15095552579}{2470477409}a^{2}-\frac{16466581969}{2470477409}a-\frac{625019752}{2470477409}$, $\frac{176377044}{2470477409}a^{11}-\frac{515107786}{2470477409}a^{10}+\frac{361454062}{2470477409}a^{9}-\frac{557628586}{2470477409}a^{8}+\frac{1243762051}{2470477409}a^{7}-\frac{12135980}{57452963}a^{6}-\frac{4789675150}{2470477409}a^{5}+\frac{4380406826}{2470477409}a^{4}-\frac{1103435606}{2470477409}a^{3}+\frac{6500508766}{2470477409}a^{2}-\frac{7288493257}{2470477409}a+\frac{497902463}{2470477409}$, $\frac{260823243}{2470477409}a^{11}-\frac{791653545}{2470477409}a^{10}+\frac{459217953}{2470477409}a^{9}-\frac{628869271}{2470477409}a^{8}+\frac{2159519766}{2470477409}a^{7}-\frac{16474824}{57452963}a^{6}-\frac{7761861922}{2470477409}a^{5}+\frac{7062005704}{2470477409}a^{4}+\frac{1749629475}{2470477409}a^{3}+\frac{11064582925}{2470477409}a^{2}-\frac{15312201134}{2470477409}a-\frac{794662201}{2470477409}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 92.1299826393 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 92.1299826393 \cdot 1}{2\cdot\sqrt{27825593350009}}\cr\approx \mathstrut & 0.217765076801 \end{aligned}\]
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
\(\Q(\sqrt{13}) \), \(\Q(\zeta_{7})^+\), 6.6.5274997.1, 6.2.31213.1, 6.2.405769.1 |
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: | 6.2.31213.1 |
Degree 8 sibling: | 8.0.68574961.1 |
Degree 12 sibling: | 12.0.164648481361.1 |
Minimal sibling: | 6.2.31213.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
\(13\) | 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
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 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |