Normalized defining polynomial
\( x^{8} - 2x^{7} - 72x^{6} - 104x^{5} + 734x^{4} + 1208x^{3} - 1712x^{2} - 1156x + 1164 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[8, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(58769636260854016\) \(\medspace = 2^{8}\cdot 19^{6}\cdot 47^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(124.78\) | 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))
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Galois root discriminant: | $2^{7/6}19^{3/4}47^{1/2}\approx 140.06053834346$ | ||
Ramified primes: | \(2\), \(19\), \(47\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{1259578}a^{7}+\frac{114734}{629789}a^{6}-\frac{6613}{1259578}a^{5}+\frac{153138}{629789}a^{4}+\frac{240394}{629789}a^{3}-\frac{6726}{629789}a^{2}+\frac{196763}{629789}a-\frac{257745}{629789}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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)
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Fundamental units: | $\frac{7564}{629789}a^{7}-\frac{2532}{629789}a^{6}-\frac{1164591}{1259578}a^{5}-\frac{1581645}{629789}a^{4}+\frac{4687269}{629789}a^{3}+\frac{15389826}{629789}a^{2}-\frac{1011728}{629789}a-\frac{11478863}{629789}$, $\frac{34409}{629789}a^{7}-\frac{430351}{1259578}a^{6}-\frac{3534721}{1259578}a^{5}+\frac{4800070}{629789}a^{4}+\frac{17141143}{629789}a^{3}-\frac{34613348}{629789}a^{2}-\frac{20410403}{629789}a+\frac{29482069}{629789}$, $\frac{22822}{629789}a^{7}-\frac{183467}{1259578}a^{6}-\frac{2693997}{1259578}a^{5}-\frac{197239}{629789}a^{4}+\frac{11066191}{629789}a^{3}+\frac{9782323}{629789}a^{2}-\frac{8557814}{629789}a-\frac{8241517}{629789}$, $\frac{46959}{629789}a^{7}-\frac{101978}{629789}a^{6}-\frac{3202835}{629789}a^{5}-\frac{5115021}{629789}a^{4}+\frac{26468969}{629789}a^{3}+\frac{58556276}{629789}a^{2}-\frac{16055918}{629789}a-\frac{48818659}{629789}$, $\frac{26205}{629789}a^{7}-\frac{16432}{629789}a^{6}-\frac{1991057}{629789}a^{5}-\frac{5106748}{629789}a^{4}+\frac{17124898}{629789}a^{3}+\frac{53074456}{629789}a^{2}-\frac{16820559}{629789}a-\frac{52973465}{629789}$, $\frac{293130}{629789}a^{7}-\frac{688821}{1259578}a^{6}-\frac{21390974}{629789}a^{5}-\frac{48121190}{629789}a^{4}+\frac{175544940}{629789}a^{3}+\frac{498717057}{629789}a^{2}-\frac{91085632}{629789}a-\frac{412820725}{629789}$, $\frac{129489}{1259578}a^{7}-\frac{492957}{1259578}a^{6}-\frac{3992487}{629789}a^{5}-\frac{479761}{629789}a^{4}+\frac{35065947}{629789}a^{3}+\frac{19578632}{629789}a^{2}-\frac{27810393}{629789}a-\frac{13959397}{629789}$ (assuming GRH) | 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: | \( 424963.918896 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 424963.918896 \cdot 2}{2\cdot\sqrt{58769636260854016}}\cr\approx \mathstrut & 0.448761428950 \end{aligned}\] (assuming GRH)
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: | 7.7.18424261696.2, 7.7.18424261696.1 |
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, deg 42, deg 42 |
Minimal sibling: | 7.7.18424261696.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.11 | $x^{8} + 4 x^{7} + 14 x^{6} + 32 x^{5} + 55 x^{4} + 60 x^{3} + 36 x^{2} + 18 x + 9$ | $4$ | $2$ | $8$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ |
\(19\) | 19.8.6.2 | $x^{8} + 72 x^{7} + 1952 x^{6} + 23760 x^{5} + 112814 x^{4} + 48888 x^{3} + 44288 x^{2} + 435600 x + 1945825$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ |
\(47\) | 47.4.2.2 | $x^{4} - 85305 x^{3} - 42072520 x^{2} - 585385 x + 11045$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
47.4.2.2 | $x^{4} - 85305 x^{3} - 42072520 x^{2} - 585385 x + 11045$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |