Normalized defining polynomial
\( x^{12} + 7x^{10} + 41x^{8} + 115x^{6} + 253x^{4} + 529x^{2} + 529 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(320761795710976\) \(\medspace = 2^{12}\cdot 23^{8}\) | 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: | \(16.18\) | 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: | $2^{7/4}23^{3/4}\approx 35.32631762433002$ | ||
Ramified primes: | \(2\), \(23\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{161}a^{8}-\frac{16}{161}a^{6}-\frac{51}{161}a^{4}-\frac{2}{7}a^{2}-\frac{1}{7}$, $\frac{1}{161}a^{9}-\frac{16}{161}a^{7}-\frac{51}{161}a^{5}-\frac{2}{7}a^{3}-\frac{1}{7}a$, $\frac{1}{44275}a^{10}-\frac{87}{44275}a^{8}-\frac{2283}{6325}a^{6}-\frac{14296}{44275}a^{4}+\frac{449}{1925}a^{2}+\frac{442}{1925}$, $\frac{1}{44275}a^{11}-\frac{87}{44275}a^{9}-\frac{2283}{6325}a^{7}-\frac{14296}{44275}a^{5}+\frac{449}{1925}a^{3}+\frac{442}{1925}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $5$ | 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: | $\frac{188}{44275}a^{10}+\frac{419}{44275}a^{8}+\frac{3522}{44275}a^{6}-\frac{51}{1925}a^{4}+\frac{116}{275}a^{2}+\frac{871}{1925}$, $\frac{15}{1771}a^{10}+\frac{10}{253}a^{8}+\frac{393}{1771}a^{6}+\frac{566}{1771}a^{4}+\frac{58}{77}a^{2}+\frac{173}{77}$, $\frac{468}{44275}a^{11}+\frac{166}{44275}a^{10}+\frac{2734}{44275}a^{9}+\frac{1783}{44275}a^{8}+\frac{16567}{44275}a^{7}+\frac{9704}{44275}a^{6}+\frac{5296}{6325}a^{5}+\frac{1368}{1925}a^{4}+\frac{3882}{1925}a^{3}+\frac{237}{275}a^{2}+\frac{5556}{1925}a+\frac{3247}{1925}$, $\frac{314}{44275}a^{11}+\frac{542}{44275}a^{10}+\frac{1557}{44275}a^{9}+\frac{2621}{44275}a^{8}+\frac{1438}{6325}a^{7}+\frac{16748}{44275}a^{6}+\frac{15556}{44275}a^{5}+\frac{1266}{1925}a^{4}+\frac{2386}{1925}a^{3}+\frac{469}{275}a^{2}+\frac{2113}{1925}a+\frac{8839}{1925}$, $\frac{1384}{44275}a^{11}+\frac{9}{161}a^{10}+\frac{8017}{44275}a^{9}+\frac{6}{23}a^{8}+\frac{45921}{44275}a^{7}+\frac{268}{161}a^{6}+\frac{96836}{44275}a^{5}+\frac{404}{161}a^{4}+\frac{8441}{1925}a^{3}+\frac{53}{7}a^{2}+\frac{19378}{1925}a+\frac{80}{7}$ | 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: | \( 88.1821625089 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 88.1821625089 \cdot 2}{2\cdot\sqrt{320761795710976}}\cr\approx \mathstrut & 0.302948381767 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 12T21):
A solvable group of order 48 |
The 10 conjugacy class representatives for $C_2\times S_4$ |
Character table for $C_2\times S_4$ |
Intermediate fields
\(\Q(\sqrt{-23}) \), 3.1.23.1 x3, 6.0.17909824.3, 6.2.778688.1, 6.0.12167.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 siblings: | 6.2.778688.1, 6.0.17909824.3 |
Degree 8 siblings: | 8.4.2425420005376.1, 8.0.2425420005376.9 |
Degree 12 siblings: | data not computed |
Degree 16 sibling: | data not computed |
Degree 24 siblings: | data not computed |
Minimal sibling: | 6.2.778688.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{6}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.6.4 | $x^{6} - 4 x^{5} + 14 x^{4} - 24 x^{3} + 100 x^{2} + 48 x + 88$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ |
2.6.6.4 | $x^{6} - 4 x^{5} + 14 x^{4} - 24 x^{3} + 100 x^{2} + 48 x + 88$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ | |
\(23\) | 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.4.3.1 | $x^{4} + 23$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
23.4.3.1 | $x^{4} + 23$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |