Normalized defining polynomial
\( x^{8} - x^{7} - 22x^{6} + 16x^{5} + 156x^{4} - 61x^{3} - 422x^{2} + 30x + 319 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[8, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(2012088347361\) \(\medspace = 3^{4}\cdot 397^{4}\) | 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: | \(34.51\) | 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: | $3^{1/2}397^{2/3}\approx 93.55944391932883$ | ||
Ramified primes: | \(3\), \(397\) | 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)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{824}a^{7}-\frac{3}{412}a^{6}+\frac{1}{103}a^{5}-\frac{3}{103}a^{4}+\frac{69}{206}a^{3}+\frac{207}{824}a^{2}+\frac{191}{824}a-\frac{101}{824}$
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: | $\frac{111}{824}a^{7}-\frac{127}{412}a^{6}-\frac{499}{206}a^{5}+\frac{1085}{206}a^{4}+\frac{1203}{103}a^{3}-\frac{19459}{824}a^{2}-\frac{12583}{824}a+\frac{22161}{824}$, $\frac{5}{103}a^{7}-\frac{17}{412}a^{6}-\frac{355}{412}a^{5}+\frac{241}{412}a^{4}+\frac{1709}{412}a^{3}-\frac{907}{412}a^{2}-\frac{1077}{206}a+\frac{967}{412}$, $\frac{1}{103}a^{7}+\frac{79}{412}a^{6}-\frac{71}{412}a^{5}-\frac{1435}{412}a^{4}+\frac{177}{412}a^{3}+\frac{6493}{412}a^{2}+\frac{279}{206}a-\frac{6069}{412}$, $\frac{17}{412}a^{7}+\frac{1}{412}a^{6}-\frac{379}{412}a^{5}-\frac{99}{412}a^{4}+\frac{2529}{412}a^{3}+\frac{339}{103}a^{2}-\frac{5199}{412}a-\frac{2249}{206}$, $\frac{1}{824}a^{7}-\frac{3}{412}a^{6}+\frac{1}{103}a^{5}-\frac{3}{103}a^{4}-\frac{137}{206}a^{3}+\frac{1031}{824}a^{2}+\frac{2663}{824}a-\frac{3397}{824}$, $\frac{71}{824}a^{7}-\frac{55}{206}a^{6}-\frac{643}{412}a^{5}+\frac{1929}{412}a^{4}+\frac{3103}{412}a^{3}-\frac{17645}{824}a^{2}-\frac{7451}{824}a+\frac{17755}{824}$, $\frac{65}{412}a^{7}+\frac{125}{412}a^{6}-\frac{1231}{412}a^{5}-\frac{2487}{412}a^{4}+\frac{5889}{412}a^{3}+\frac{3132}{103}a^{2}-\frac{4683}{412}a-\frac{5703}{206}$ | 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: | \( 4575.92530631 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 4575.92530631 \cdot 1}{2\cdot\sqrt{2012088347361}}\cr\approx \mathstrut & 0.412919481620 \end{aligned}\]
Galois group
$\SL(2,3)$ (as 8T12):
A solvable group of order 24 |
The 7 conjugacy class representatives for $\SL(2,3)$ |
Character table for $\SL(2,3)$ |
Intermediate fields
4.4.157609.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
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 | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(397\) | $\Q_{397}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{397}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $3$ | $3$ | $1$ | $2$ | ||||
Deg $3$ | $3$ | $1$ | $2$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.397.3t1.a.a | $1$ | $ 397 $ | 3.3.157609.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.397.3t1.a.b | $1$ | $ 397 $ | 3.3.157609.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
2.1418481.24t7.a.a | $2$ | $ 3^{2} \cdot 397^{2}$ | 8.8.2012088347361.1 | $\SL(2,3)$ (as 8T12) | $-1$ | $2$ | |
* | 2.3573.8t12.a.a | $2$ | $ 3^{2} \cdot 397 $ | 8.8.2012088347361.1 | $\SL(2,3)$ (as 8T12) | $0$ | $2$ |
* | 2.3573.8t12.a.b | $2$ | $ 3^{2} \cdot 397 $ | 8.8.2012088347361.1 | $\SL(2,3)$ (as 8T12) | $0$ | $2$ |
* | 3.157609.4t4.a.a | $3$ | $ 397^{2}$ | 4.4.157609.1 | $A_4$ (as 4T4) | $1$ | $3$ |