Normalized defining polynomial
\( x^{8} - 3x^{7} + 109x^{6} + 138x^{5} + 3801x^{4} + 13938x^{3} + 54538x^{2} + 67350x + 58153 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(12696463968316569\) \(\medspace = 3^{6}\cdot 7^{6}\cdot 23^{6}\) | 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: | \(103.03\) | 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: | $3^{3/4}7^{3/4}23^{3/4}\approx 103.02920484662951$ | ||
Ramified primes: | \(3\), \(7\), \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | 8.0.12696463968316569.1$^{8}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{2}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{6}+\frac{1}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{31119818192615}a^{7}-\frac{255828105610}{6223963638523}a^{6}+\frac{1097369143347}{31119818192615}a^{5}-\frac{13951143330293}{31119818192615}a^{4}+\frac{7834370920704}{31119818192615}a^{3}+\frac{2905252260572}{31119818192615}a^{2}+\frac{15392832777473}{31119818192615}a-\frac{4261686154764}{31119818192615}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{6}\times C_{6}$, which has order $72$
Unit group
Rank: | $3$ | 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: | $\frac{12771}{916987895}a^{7}-\frac{38113}{183397579}a^{6}+\frac{2314013}{916987895}a^{5}-\frac{3335389}{183397579}a^{4}+\frac{67523884}{916987895}a^{3}-\frac{338796829}{916987895}a^{2}-\frac{512754586}{916987895}a-\frac{530098883}{916987895}$, $\frac{155340656}{31119818192615}a^{7}+\frac{365756680}{6223963638523}a^{6}-\frac{18079249117}{31119818192615}a^{5}+\frac{77113557520}{6223963638523}a^{4}-\frac{1206563784486}{31119818192615}a^{3}+\frac{6069869597776}{31119818192615}a^{2}+\frac{9131842645924}{31119818192615}a+\frac{26704996591462}{31119818192615}$, $\frac{4005433591}{31119818192615}a^{7}+\frac{1833397763}{6223963638523}a^{6}+\frac{336653509833}{31119818192615}a^{5}+\frac{330894068375}{6223963638523}a^{4}+\frac{6020112389884}{31119818192615}a^{3}+\frac{9353990738231}{31119818192615}a^{2}+\frac{8868410494894}{31119818192615}a+\frac{2929776989572}{31119818192615}$ | 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: | \( 405.962625769 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 405.962625769 \cdot 72}{2\cdot\sqrt{12696463968316569}}\cr\approx \mathstrut & 0.202146690152 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $Q_8$ |
Character table for $Q_8$ |
Intermediate fields
\(\Q(\sqrt{21}) \), \(\Q(\sqrt{69}) \), \(\Q(\sqrt{161}) \), \(\Q(\sqrt{21}, \sqrt{69})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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 | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
\(7\) | 7.8.6.1 | $x^{8} + 14 x^{4} - 245$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
\(23\) | 23.8.6.1 | $x^{8} - 138 x^{4} - 217948$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.69.2t1.a.a | $1$ | $ 3 \cdot 23 $ | \(\Q(\sqrt{69}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.161.2t1.a.a | $1$ | $ 7 \cdot 23 $ | \(\Q(\sqrt{161}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.21.2t1.a.a | $1$ | $ 3 \cdot 7 $ | \(\Q(\sqrt{21}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
*2 | 2.233289.8t5.b.a | $2$ | $ 3^{2} \cdot 7^{2} \cdot 23^{2}$ | 8.0.12696463968316569.1 | $Q_8$ (as 8T5) | $-1$ | $-2$ |