Normalized defining polynomial
\( x^{8} - x^{7} - 49x^{6} - 16x^{5} + 511x^{4} + 367x^{3} - 1499x^{2} - 798x + 1372 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[8, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(235260548044817\) \(\medspace = 113^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(62.58\) | 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: | $113^{7/8}\approx 62.58116419486453$ | ||
Ramified primes: | \(113\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{113}) \) | ||
$\card{ \Gal(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(113\) | ||
Dirichlet character group: | $\lbrace$$\chi_{113}(1,·)$, $\chi_{113}(98,·)$, $\chi_{113}(69,·)$, $\chi_{113}(44,·)$, $\chi_{113}(15,·)$, $\chi_{113}(112,·)$, $\chi_{113}(18,·)$, $\chi_{113}(95,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{14}a^{5}+\frac{1}{7}a^{3}-\frac{1}{2}a^{2}+\frac{2}{7}a$, $\frac{1}{56}a^{6}-\frac{1}{56}a^{5}-\frac{3}{14}a^{4}+\frac{19}{56}a^{3}+\frac{11}{56}a^{2}-\frac{9}{28}a-\frac{1}{2}$, $\frac{1}{9296}a^{7}-\frac{13}{4648}a^{6}+\frac{269}{9296}a^{5}+\frac{2223}{9296}a^{4}+\frac{3}{581}a^{3}+\frac{827}{9296}a^{2}-\frac{1293}{4648}a-\frac{91}{332}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$, $7$ |
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)
| |
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: | $\frac{20}{581}a^{7}-\frac{337}{2324}a^{6}-\frac{3131}{2324}a^{5}+\frac{4675}{1162}a^{4}+\frac{23677}{2324}a^{3}-\frac{51783}{2324}a^{2}-\frac{13042}{581}a+\frac{2431}{83}$, $\frac{71}{9296}a^{7}-\frac{5}{2324}a^{6}-\frac{473}{1328}a^{5}-\frac{3519}{9296}a^{4}+\frac{1773}{664}a^{3}+\frac{36971}{9296}a^{2}-\frac{9965}{4648}a-\frac{983}{332}$, $\frac{71}{4648}a^{7}-\frac{103}{4648}a^{6}-\frac{807}{1162}a^{5}-\frac{199}{4648}a^{4}+\frac{27893}{4648}a^{3}+\frac{11057}{2324}a^{2}-\frac{6081}{581}a-\frac{865}{83}$, $\frac{629}{9296}a^{7}-\frac{1039}{4648}a^{6}-\frac{26347}{9296}a^{5}+\frac{51011}{9296}a^{4}+\frac{54443}{2324}a^{3}-\frac{266325}{9296}a^{2}-\frac{209721}{4648}a+\frac{15137}{332}$, $\frac{447}{2324}a^{7}-\frac{1413}{2324}a^{6}-\frac{9391}{1162}a^{5}+\frac{33371}{2324}a^{4}+\frac{153011}{2324}a^{3}-\frac{41607}{581}a^{2}-\frac{142553}{1162}a+\frac{9621}{83}$, $\frac{789}{9296}a^{7}-\frac{261}{1162}a^{6}-\frac{35265}{9296}a^{5}+\frac{45475}{9296}a^{4}+\frac{164767}{4648}a^{3}-\frac{255683}{9296}a^{2}-\frac{384563}{4648}a+\frac{23319}{332}$, $\frac{417}{4648}a^{7}-\frac{209}{664}a^{6}-\frac{8363}{2324}a^{5}+\frac{5129}{664}a^{4}+\frac{121857}{4648}a^{3}-\frac{6409}{166}a^{2}-\frac{55409}{1162}a+\frac{4474}{83}$ | 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: | \( 172951.810041 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 172951.810041 \cdot 1}{2\cdot\sqrt{235260548044817}}\cr\approx \mathstrut & 1.44331301868 \end{aligned}\]
Galois group
A cyclic group of order 8 |
The 8 conjugacy class representatives for $C_8$ |
Character table for $C_8$ |
Intermediate fields
\(\Q(\sqrt{113}) \), 4.4.1442897.1 |
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 | ${\href{/padicField/2.2.0.1}{2} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.1.0.1}{1} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
---|---|---|---|---|---|---|---|
\(113\) | 113.8.7.1 | $x^{8} + 113$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.113.8t1.a.a | $1$ | $ 113 $ | 8.8.235260548044817.1 | $C_8$ (as 8T1) | $0$ | $1$ |
* | 1.113.4t1.a.a | $1$ | $ 113 $ | 4.4.1442897.1 | $C_4$ (as 4T1) | $0$ | $1$ |
* | 1.113.8t1.a.b | $1$ | $ 113 $ | 8.8.235260548044817.1 | $C_8$ (as 8T1) | $0$ | $1$ |
* | 1.113.2t1.a.a | $1$ | $ 113 $ | \(\Q(\sqrt{113}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.113.8t1.a.c | $1$ | $ 113 $ | 8.8.235260548044817.1 | $C_8$ (as 8T1) | $0$ | $1$ |
* | 1.113.4t1.a.b | $1$ | $ 113 $ | 4.4.1442897.1 | $C_4$ (as 4T1) | $0$ | $1$ |
* | 1.113.8t1.a.d | $1$ | $ 113 $ | 8.8.235260548044817.1 | $C_8$ (as 8T1) | $0$ | $1$ |