Normalized defining polynomial
\( x^{8} - x^{7} - 9x^{6} + 8x^{5} + 25x^{4} - 24x^{3} - 18x^{2} + 30x - 3 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(6512974209\) \(\medspace = 3^{6}\cdot 7^{4}\cdot 61^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.85\) | 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: | $3^{3/4}7^{1/2}61^{1/2}\approx 47.1036844046804$ | ||
Ramified primes: | \(3\), \(7\), \(61\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{4}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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)
| |
Fundamental units: | $\frac{1}{3}a^{6}-\frac{2}{3}a^{5}-\frac{5}{3}a^{4}+4a^{3}+a^{2}-6a+3$, $\frac{1}{3}a^{6}-\frac{2}{3}a^{5}-\frac{5}{3}a^{4}+3a^{3}+2a^{2}-3a+1$, $\frac{1}{3}a^{7}-a^{6}-a^{5}+\frac{17}{3}a^{4}-4a^{3}-7a^{2}+13a-4$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{5}{3}a^{4}-2a^{3}+1$, $\frac{2}{3}a^{7}-\frac{4}{3}a^{6}-\frac{10}{3}a^{5}+8a^{4}+a^{3}-12a^{2}+10a-1$ | 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: | \( 127.427485202 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 127.427485202 \cdot 1}{2\cdot\sqrt{6512974209}}\cr\approx \mathstrut & 0.498681384839 \end{aligned}\]
Galois group
$C_2\wr C_2^2$ (as 8T29):
A solvable group of order 64 |
The 16 conjugacy class representatives for $(((C_4 \times C_2): C_2):C_2):C_2$ |
Character table for $(((C_4 \times C_2): C_2):C_2):C_2$ |
Intermediate fields
\(\Q(\sqrt{21}) \), 4.2.1323.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 32 siblings: | data not computed |
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 | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/5.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.4.3.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
3.4.3.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
\(7\) | 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(61\) | 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.61.2t1.a.a | $1$ | $ 61 $ | \(\Q(\sqrt{61}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.427.2t1.a.a | $1$ | $ 7 \cdot 61 $ | \(\Q(\sqrt{-427}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.183.2t1.a.a | $1$ | $ 3 \cdot 61 $ | \(\Q(\sqrt{-183}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.21.2t1.a.a | $1$ | $ 3 \cdot 7 $ | \(\Q(\sqrt{21}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.1281.2t1.a.a | $1$ | $ 3 \cdot 7 \cdot 61 $ | \(\Q(\sqrt{1281}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.8967.4t3.a.a | $2$ | $ 3 \cdot 7^{2} \cdot 61 $ | 4.2.546987.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.3843.4t3.d.a | $2$ | $ 3^{2} \cdot 7 \cdot 61 $ | 4.2.4922883.2 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.183.4t3.b.a | $2$ | $ 3 \cdot 61 $ | 4.2.11163.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.234423.4t3.a.a | $2$ | $ 3^{2} \cdot 7 \cdot 61^{2}$ | 4.2.4922883.3 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.3843.4t3.c.a | $2$ | $ 3^{2} \cdot 7 \cdot 61 $ | 4.2.4922883.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 2.63.4t3.c.a | $2$ | $ 3^{2} \cdot 7 $ | 4.2.1323.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
4.4922883.8t29.a.a | $4$ | $ 3^{3} \cdot 7^{2} \cdot 61^{2}$ | 8.4.6512974209.1 | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $-2$ | |
* | 4.4922883.8t29.b.a | $4$ | $ 3^{3} \cdot 7^{2} \cdot 61^{2}$ | 8.4.6512974209.1 | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $2$ |