Normalized defining polynomial
\( x^{8} - 2x^{7} - 2x^{6} + 7x^{5} + 3x^{4} - 34x^{3} + 93x^{2} - 29x + 39 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(591559418641\) \(\medspace = 877^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(29.61\) | 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: | $877^{2/3}\approx 91.62199182116717$ | ||
Ramified primes: | \(877\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{124000}a^{7}-\frac{8399}{24800}a^{6}-\frac{31967}{124000}a^{5}-\frac{16881}{62000}a^{4}-\frac{48331}{124000}a^{3}+\frac{55649}{124000}a^{2}+\frac{8909}{31000}a-\frac{30577}{124000}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $3$ | 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: | $\frac{2473}{62000}a^{7}-\frac{727}{12400}a^{6}-\frac{4391}{62000}a^{5}+\frac{10287}{31000}a^{4}+\frac{13437}{62000}a^{3}-\frac{82023}{62000}a^{2}+\frac{52957}{15500}a-\frac{38921}{62000}$, $\frac{17311}{31000}a^{7}-\frac{5089}{6200}a^{6}-\frac{61737}{31000}a^{5}+\frac{56509}{15500}a^{4}+\frac{156059}{31000}a^{3}-\frac{605161}{31000}a^{2}+\frac{293199}{7750}a+\frac{595553}{31000}$, $\frac{7899}{124000}a^{7}-\frac{3701}{24800}a^{6}-\frac{43333}{124000}a^{5}+\frac{18981}{62000}a^{4}+\frac{29431}{124000}a^{3}-\frac{504549}{124000}a^{2}+\frac{33191}{31000}a-\frac{223723}{124000}$ | 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: | \( 426.628363544 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 426.628363544 \cdot 2}{2\cdot\sqrt{591559418641}}\cr\approx \mathstrut & 0.864509981595 \end{aligned}\]
Galois group
A solvable group of order 168 |
The 8 conjugacy class representatives for $C_2^3:(C_7: C_3)$ |
Character table for $C_2^3:(C_7: C_3)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 14 sibling: | deg 14 |
Degree 24 sibling: | deg 24 |
Degree 28 sibling: | deg 28 |
Degree 42 sibling: | 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 | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(877\) | $\Q_{877}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{877}$ | $x$ | $1$ | $1$ | $0$ | 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 | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.877.3t1.a.a | $1$ | $ 877 $ | 3.3.769129.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.877.3t1.a.b | $1$ | $ 877 $ | 3.3.769129.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
3.769129.7t3.a.a | $3$ | $ 877^{2}$ | 7.7.591559418641.1 | $C_7:C_3$ (as 7T3) | $0$ | $3$ | |
3.769129.7t3.a.b | $3$ | $ 877^{2}$ | 7.7.591559418641.1 | $C_7:C_3$ (as 7T3) | $0$ | $3$ | |
* | 7.591559418641.8t36.b.a | $7$ | $ 877^{4}$ | 8.0.591559418641.2 | $C_2^3:(C_7: C_3)$ (as 8T36) | $1$ | $-1$ |
7.518...157.24t283.b.a | $7$ | $ 877^{5}$ | 8.0.591559418641.2 | $C_2^3:(C_7: C_3)$ (as 8T36) | $0$ | $-1$ | |
7.518...157.24t283.b.b | $7$ | $ 877^{5}$ | 8.0.591559418641.2 | $C_2^3:(C_7: C_3)$ (as 8T36) | $0$ | $-1$ |