Normalized defining polynomial
\( x^{8} - 2x^{7} + 16x^{6} - 20x^{5} + 56x^{4} - 56x^{3} + 80x^{2} - 80x + 64 \)
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: | \(1135929640000\) \(\medspace = 2^{6}\cdot 5^{4}\cdot 73^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(32.13\) | 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: | $2^{6/7}5^{1/2}73^{2/3}\approx 70.74991995054486$ | ||
Ramified primes: | \(2\), \(5\), \(73\) | 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{80}a^{7}+\frac{1}{40}a^{6}+\frac{1}{20}a^{5}-\frac{1}{20}a^{4}-\frac{1}{5}a^{2}+\frac{1}{5}a-\frac{1}{5}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}$, which has order $4$
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: | $\frac{7}{40}a^{7}+\frac{1}{10}a^{6}+\frac{49}{20}a^{5}+\frac{61}{20}a^{4}+9a^{3}+\frac{56}{5}a^{2}+\frac{69}{5}a+\frac{51}{5}$, $\frac{1}{40}a^{7}-\frac{1}{5}a^{6}+\frac{17}{20}a^{5}-\frac{57}{20}a^{4}+\frac{11}{2}a^{3}-\frac{32}{5}a^{2}+\frac{22}{5}a+\frac{3}{5}$, $\frac{1}{2}a^{7}-a^{6}+\frac{15}{2}a^{5}-8a^{4}+18a^{3}-4a^{2}-2a+9$ | 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: | \( 689.315412453 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 689.315412453 \cdot 4}{2\cdot\sqrt{1135929640000}}\cr\approx \mathstrut & 2.01600563746 \end{aligned}\]
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 | ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
\(5\) | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(73\) | 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
73.6.4.1 | $x^{6} + 210 x^{5} + 14715 x^{4} + 345246 x^{3} + 88905 x^{2} + 1076160 x + 24967804$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.73.3t1.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) | $0$ | $3$ | |
* | 7.113...000.8t36.a.a | $7$ | $ 2^{6} \cdot 5^{4} \cdot 73^{4}$ | 8.0.1135929640000.1 | $C_2^3:(C_7: C_3)$ (as 8T36) | $1$ | $-1$ |
7.829...000.24t283.a.a | $7$ | $ 2^{6} \cdot 5^{4} \cdot 73^{5}$ | 8.0.1135929640000.1 | $C_2^3:(C_7: C_3)$ (as 8T36) | $0$ | $-1$ | |
7.829...000.24t283.a.b | $7$ | $ 2^{6} \cdot 5^{4} \cdot 73^{5}$ | 8.0.1135929640000.1 | $C_2^3:(C_7: C_3)$ (as 8T36) | $0$ | $-1$ |