Normalized defining polynomial
\( x^{8} - 3x^{7} + 29x^{6} - 113x^{5} + 383x^{4} - 1100x^{3} + 2424x^{2} - 3272x + 5488 \)
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: | \(199390751295241\) \(\medspace = 11^{6}\cdot 103^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(61.30\) | 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: | $11^{6/7}103^{1/2}\approx 79.25759937842756$ | ||
Ramified primes: | \(11\), \(103\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{11}a^{4}+\frac{3}{11}a^{3}-\frac{5}{11}a^{2}-\frac{5}{11}a-\frac{4}{11}$, $\frac{1}{22}a^{5}-\frac{1}{22}a^{4}+\frac{5}{22}a^{3}-\frac{7}{22}a^{2}+\frac{5}{22}a-\frac{3}{11}$, $\frac{1}{484}a^{6}-\frac{1}{44}a^{5}+\frac{13}{484}a^{4}-\frac{41}{484}a^{3}-\frac{157}{484}a^{2}+\frac{16}{121}a-\frac{38}{121}$, $\frac{1}{968}a^{7}-\frac{1}{968}a^{6}-\frac{9}{968}a^{5}+\frac{1}{968}a^{4}+\frac{357}{968}a^{3}+\frac{149}{484}a^{2}-\frac{5}{121}a-\frac{14}{121}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{12}$, which has order $12$
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: | $\frac{45}{968}a^{7}-\frac{63}{968}a^{6}+\frac{717}{968}a^{5}-\frac{2257}{968}a^{4}+\frac{2503}{968}a^{3}+2a^{2}-\frac{2509}{242}a+\frac{8545}{121}$, $\frac{1}{22}a^{7}-\frac{157}{484}a^{6}+\frac{71}{44}a^{5}-\frac{4043}{484}a^{4}+\frac{13543}{484}a^{3}-\frac{31583}{484}a^{2}+\frac{29087}{242}a-\frac{16474}{121}$, $\frac{225}{484}a^{7}-\frac{67}{242}a^{6}+\frac{965}{121}a^{5}-\frac{463}{22}a^{4}+\frac{2445}{121}a^{3}-\frac{16911}{484}a^{2}-\frac{1401}{242}a+\frac{105984}{121}$ | 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: | \( 2081.22156701 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 2081.22156701 \cdot 12}{2\cdot\sqrt{199390751295241}}\cr\approx \mathstrut & 1.37827708793 \end{aligned}\]
Galois group
$\GL(3,2)$ (as 8T37):
A non-solvable group of order 168 |
The 6 conjugacy class representatives for $\PSL(2,7)$ |
Character table for $\PSL(2,7)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 7 siblings: | 7.3.18794490649.1, 7.3.18794490649.2 |
Degree 14 siblings: | deg 14, deg 14 |
Degree 21 sibling: | deg 21 |
Degree 24 sibling: | deg 24 |
Degree 28 sibling: | deg 28 |
Degree 42 siblings: | deg 42, some data not computed |
Minimal sibling: | 7.3.18794490649.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
11.7.6.1 | $x^{7} + 11$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
\(103\) | 103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
103.2.1.2 | $x^{2} + 103$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |