Normalized defining polynomial
\( x^{8} - 2x^{7} + 10x^{6} - 6x^{5} + 2x^{4} + 8x^{3} - 16x^{2} + 16x + 26 \)
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: | \(116319195136\) \(\medspace = 2^{12}\cdot 73^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(24.17\) | 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: | $2^{45/28}73^{2/3}\approx 53.21247356032676$ | ||
Ramified primes: | \(2\), \(73\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{13283}a^{7}+\frac{3213}{13283}a^{6}-\frac{4369}{13283}a^{5}-\frac{6210}{13283}a^{4}-\frac{799}{13283}a^{3}-\frac{5158}{13283}a^{2}-\frac{5802}{13283}a-\frac{4082}{13283}$
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: | $\frac{33}{13283}a^{7}-\frac{235}{13283}a^{6}+\frac{1936}{13283}a^{5}-\frac{5685}{13283}a^{4}+\frac{13482}{13283}a^{3}-\frac{10818}{13283}a^{2}-\frac{5504}{13283}a+\frac{11407}{13283}$, $\frac{285}{13283}a^{7}-\frac{822}{13283}a^{6}+\frac{3437}{13283}a^{5}-\frac{3211}{13283}a^{4}-\frac{1904}{13283}a^{3}+\frac{4383}{13283}a^{2}-\frac{6478}{13283}a-\frac{7749}{13283}$, $\frac{335}{13283}a^{7}+\frac{432}{13283}a^{6}-\frac{2485}{13283}a^{5}+\frac{18364}{13283}a^{4}-\frac{41854}{13283}a^{3}+\frac{65275}{13283}a^{2}-\frac{70767}{13283}a+\frac{53811}{13283}$ | 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: | \( 229.398855642 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 229.398855642 \cdot 2}{2\cdot\sqrt{116319195136}}\cr\approx \mathstrut & 1.04829864948 \end{aligned}\]
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: | deg 42 |
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 | R | ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.12.27 | $x^{8} + 2 x^{5} + 2 x^{4} + 2$ | $8$ | $1$ | $12$ | $C_2^3:(C_7: C_3)$ | $[12/7, 12/7, 12/7]_{7}^{3}$ |
\(73\) | $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.73.3t1.a.a | $1$ | $ 73 $ | 3.3.5329.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.73.3t1.a.b | $1$ | $ 73 $ | 3.3.5329.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
3.42632.7t3.a.a | $3$ | $ 2^{3} \cdot 73^{2}$ | 7.7.1817487424.1 | $C_7:C_3$ (as 7T3) | $0$ | $3$ | |
3.42632.7t3.a.b | $3$ | $ 2^{3} \cdot 73^{2}$ | 7.7.1817487424.1 | $C_7:C_3$ (as 7T3) | $0$ | $3$ | |
* | 7.116319195136.8t36.b.a | $7$ | $ 2^{12} \cdot 73^{4}$ | 8.0.116319195136.3 | $C_2^3:(C_7: C_3)$ (as 8T36) | $1$ | $-1$ |
7.849...928.24t283.b.a | $7$ | $ 2^{12} \cdot 73^{5}$ | 8.0.116319195136.3 | $C_2^3:(C_7: C_3)$ (as 8T36) | $0$ | $-1$ | |
7.849...928.24t283.b.b | $7$ | $ 2^{12} \cdot 73^{5}$ | 8.0.116319195136.3 | $C_2^3:(C_7: C_3)$ (as 8T36) | $0$ | $-1$ |