Normalized defining polynomial
\( x^{8} - x^{7} + 98x^{6} - 105x^{5} + 3191x^{4} + 1665x^{3} + 44072x^{2} + 47933x + 328171 \)
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)
| |
Discriminant: | \(752823265640625\) \(\medspace = 3^{4}\cdot 5^{6}\cdot 29^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(72.37\) | 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: | $3^{1/2}5^{3/4}29^{3/4}\approx 72.37467541740445$ | ||
Ramified primes: | \(3\), \(5\), \(29\) | 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: | 8.0.752823265640625.1$^{8}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{15}a^{6}+\frac{1}{15}a^{5}-\frac{2}{5}a^{4}+\frac{2}{15}a^{3}+\frac{2}{5}a^{2}-\frac{1}{3}a+\frac{1}{15}$, $\frac{1}{117260457787365}a^{7}+\frac{3851137782358}{117260457787365}a^{6}+\frac{1910000966747}{39086819262455}a^{5}+\frac{8897059003084}{23452091557473}a^{4}+\frac{1501423814376}{7817363852491}a^{3}-\frac{46836653002283}{117260457787365}a^{2}-\frac{53448990962894}{117260457787365}a+\frac{14673905085209}{39086819262455}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{4}$, which has order $256$
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)
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Fundamental units: | $\frac{535873}{23400610215}a^{7}-\frac{284642}{4680122043}a^{6}+\frac{44188054}{23400610215}a^{5}-\frac{141474286}{23400610215}a^{4}+\frac{985804337}{23400610215}a^{3}-\frac{715872113}{23400610215}a^{2}+\frac{3294618968}{23400610215}a+\frac{5975456947}{23400610215}$, $\frac{32870}{1768767747}a^{7}-\frac{134000}{1768767747}a^{6}+\frac{1660463}{1768767747}a^{5}-\frac{13761770}{1768767747}a^{4}-\frac{3442280}{1768767747}a^{3}-\frac{280864840}{1768767747}a^{2}-\frac{447730730}{1768767747}a-\frac{2738873311}{1768767747}$, $\frac{246136511}{117260457787365}a^{7}+\frac{99582158}{23452091557473}a^{6}+\frac{92497143848}{117260457787365}a^{5}-\frac{237317693477}{117260457787365}a^{4}+\frac{3418043445499}{117260457787365}a^{3}+\frac{3331131864989}{117260457787365}a^{2}+\frac{18694987747261}{117260457787365}a+\frac{128084296060739}{117260457787365}$ | 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: | \( 40.3281811419 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 40.3281811419 \cdot 256}{2\cdot\sqrt{752823265640625}}\cr\approx \mathstrut & 0.293218631228 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $Q_8$ |
Character table for $Q_8$ |
Intermediate fields
\(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{29})\) |
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 | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(5\) | 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(29\) | 29.4.3.1 | $x^{4} + 116$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
29.4.3.1 | $x^{4} + 116$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.29.2t1.a.a | $1$ | $ 29 $ | \(\Q(\sqrt{29}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.145.2t1.a.a | $1$ | $ 5 \cdot 29 $ | \(\Q(\sqrt{145}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
*2 | 2.189225.8t5.a.a | $2$ | $ 3^{2} \cdot 5^{2} \cdot 29^{2}$ | 8.0.752823265640625.1 | $Q_8$ (as 8T5) | $-1$ | $-2$ |