Normalized defining polynomial
\( x^{12} - 2x^{11} + 4x^{9} - 8x^{8} + 20x^{7} - 28x^{6} - 8x^{5} + 68x^{4} - 72x^{3} + 16x^{2} + 16x - 8 \)
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: | \(1511207993344\) \(\medspace = 2^{18}\cdot 7^{8}\) | 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: | \(10.35\) | 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: | $2^{7/4}7^{2/3}\approx 12.308388215503062$ | ||
Ramified primes: | \(2\), \(7\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{724}a^{11}+\frac{8}{181}a^{10}+\frac{1}{362}a^{9}+\frac{18}{181}a^{8}-\frac{47}{362}a^{7}+\frac{41}{362}a^{6}-\frac{34}{181}a^{5}+\frac{37}{362}a^{4}-\frac{78}{181}a^{3}+\frac{45}{181}a^{2}+\frac{86}{181}a+\frac{32}{181}$
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: | $\frac{237}{362}a^{11}-\frac{579}{724}a^{10}-\frac{125}{181}a^{9}+\frac{387}{181}a^{8}-\frac{641}{181}a^{7}+\frac{3687}{362}a^{6}-\frac{1817}{181}a^{5}-\frac{5087}{362}a^{4}+\frac{6287}{181}a^{3}-\frac{3467}{181}a^{2}-\frac{1228}{181}a+\frac{1231}{181}$, $\frac{17}{724}a^{11}+\frac{1}{724}a^{10}+\frac{17}{362}a^{9}-\frac{43}{724}a^{8}-\frac{75}{362}a^{7}+\frac{77}{181}a^{6}-\frac{35}{181}a^{5}+\frac{267}{362}a^{4}-\frac{59}{181}a^{3}-\frac{321}{181}a^{2}+\frac{376}{181}a+\frac{1}{181}$, $\frac{29}{181}a^{11}-\frac{135}{362}a^{10}-\frac{65}{362}a^{9}+\frac{569}{724}a^{8}-\frac{192}{181}a^{7}+\frac{568}{181}a^{6}-\frac{1553}{362}a^{5}-\frac{1319}{362}a^{4}+\frac{2355}{181}a^{3}-\frac{1477}{181}a^{2}-\frac{522}{181}a+\frac{635}{181}$, $\frac{527}{724}a^{11}-\frac{693}{724}a^{10}-\frac{575}{724}a^{9}+\frac{436}{181}a^{8}-\frac{710}{181}a^{7}+\frac{4231}{362}a^{6}-\frac{2171}{181}a^{5}-\frac{2830}{181}a^{4}+\frac{7040}{181}a^{3}-\frac{3978}{181}a^{2}-\frac{1195}{181}a+\frac{1117}{181}$, $\frac{49}{181}a^{11}-\frac{425}{724}a^{10}-\frac{151}{724}a^{9}+\frac{899}{724}a^{8}-\frac{705}{362}a^{7}+\frac{941}{181}a^{6}-\frac{2649}{362}a^{5}-\frac{899}{181}a^{4}+\frac{3717}{181}a^{3}-\frac{2764}{181}a^{2}-\frac{339}{181}a+\frac{842}{181}$, $\frac{71}{181}a^{11}-\frac{81}{181}a^{10}-\frac{337}{724}a^{9}+\frac{225}{181}a^{8}-\frac{339}{181}a^{7}+\frac{1116}{181}a^{6}-\frac{968}{181}a^{5}-\frac{1624}{181}a^{4}+\frac{3550}{181}a^{3}-\frac{1700}{181}a^{2}-\frac{735}{181}a+\frac{581}{181}$, $\frac{35}{724}a^{11}+\frac{17}{362}a^{10}-\frac{111}{724}a^{9}-\frac{7}{362}a^{8}-\frac{8}{181}a^{7}+\frac{84}{181}a^{6}+\frac{335}{362}a^{5}-\frac{348}{181}a^{4}-\frac{15}{181}a^{3}+\frac{308}{181}a^{2}-\frac{67}{181}a-\frac{147}{181}$ | 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: | \( 17.6734900062 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 17.6734900062 \cdot 1}{2\cdot\sqrt{1511207993344}}\cr\approx \mathstrut & 0.179254331206 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 12T6):
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
\(\Q(\zeta_{7})^+\), 6.4.153664.1 x2, 6.2.153664.1 |
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: | 6.4.153664.1 |
Degree 8 sibling: | 8.0.39337984.2 |
Degree 12 sibling: | 12.0.1511207993344.2 |
Minimal sibling: | 6.4.153664.1 |
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 | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.18.31 | $x^{12} - 4 x^{11} + 22 x^{10} + 16 x^{9} + 50 x^{8} + 32 x^{7} + 144 x^{6} + 96 x^{5} + 236 x^{4} + 248 x^{2} + 248$ | $4$ | $3$ | $18$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ |
\(7\) | 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 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |