Normalized defining polynomial
\( x^{8} - 28x^{6} - 112x^{5} - 210x^{4} - 224x^{3} - 140x^{2} - 48x + 823561 \)
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)
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Discriminant: | \(19501894337558159417628591379185664\) \(\medspace = 2^{20}\cdot 51473^{6}\) | 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: | \(19\,331.23\) | 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: | $2^{43/16}51473^{6/7}\approx 70390.62337700772$ | ||
Ramified primes: | \(2\), \(51473\) | 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{4}-\frac{1}{4}$, $\frac{1}{8}a^{5}-\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{3}{8}a-\frac{1}{8}$, $\frac{1}{8}a^{6}-\frac{1}{8}a^{4}-\frac{1}{8}a^{2}+\frac{1}{8}$, $\frac{1}{80}a^{7}+\frac{1}{80}a^{6}+\frac{3}{80}a^{5}-\frac{9}{80}a^{4}+\frac{11}{80}a^{3}-\frac{13}{80}a^{2}-\frac{23}{80}a+\frac{29}{80}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
Rank: | $3$ | 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)
| |
Fundamental units: | $\frac{35\!\cdots\!21}{80}a^{7}+\frac{27\!\cdots\!91}{80}a^{6}-\frac{21\!\cdots\!47}{80}a^{5}+\frac{91\!\cdots\!51}{80}a^{4}+\frac{32\!\cdots\!11}{80}a^{3}-\frac{21\!\cdots\!83}{80}a^{2}+\frac{57\!\cdots\!27}{80}a+\frac{57\!\cdots\!69}{80}$, $\frac{16\!\cdots\!81}{5}a^{7}-\frac{17\!\cdots\!97}{40}a^{6}-\frac{65\!\cdots\!11}{40}a^{5}+\frac{23\!\cdots\!57}{10}a^{4}+\frac{32\!\cdots\!89}{20}a^{3}-\frac{70\!\cdots\!49}{40}a^{2}+\frac{14\!\cdots\!21}{40}a+\frac{65\!\cdots\!91}{20}$, $\frac{65\!\cdots\!97}{10}a^{7}-\frac{30\!\cdots\!23}{10}a^{6}-\frac{22\!\cdots\!29}{10}a^{5}+\frac{60\!\cdots\!91}{5}a^{4}-\frac{93\!\cdots\!53}{10}a^{3}-\frac{15\!\cdots\!03}{5}a^{2}+\frac{18\!\cdots\!99}{10}a+\frac{38\!\cdots\!73}{10}$ (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)]
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Regulator: | \( 73355994708200 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 73355994708200 \cdot 2}{2\cdot\sqrt{19501894337558159417628591379185664}}\cr\approx \mathstrut & 0.818684719333306 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 20160 |
The 14 conjugacy class representatives for $A_8$ |
Character table for $A_8$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 15 siblings: | deg 15, deg 15 |
Degree 28 sibling: | deg 28 |
Degree 35 sibling: | deg 35 |
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 | ${\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.20.79 | $x^{8} + 4 x^{7} + 4 x^{5} + 2 x^{4} + 4 x^{2} + 2$ | $8$ | $1$ | $20$ | $C_2^4:C_6$ | $[2, 2, 2, 3, 3]^{3}$ |
\(51473\) | $\Q_{51473}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $7$ | $7$ | $1$ | $6$ |