Normalized defining polynomial
\( x^{8} - 28x^{6} - 84x^{5} + 210x^{4} + 5096x^{3} + 13552x^{2} + 6008x - 2002 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(14513641568075776\) \(\medspace = 2^{20}\cdot 7^{12}\) | 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: | \(104.77\) | 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: | $2^{5/2}7^{12/7}\approx 158.96974819747277$ | ||
Ramified primes: | \(2\), \(7\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{1930820643809}a^{7}+\frac{901176109461}{1930820643809}a^{6}+\frac{56128360725}{1930820643809}a^{5}+\frac{228144021415}{1930820643809}a^{4}+\frac{125805144511}{1930820643809}a^{3}+\frac{96291387924}{1930820643809}a^{2}-\frac{429215513051}{1930820643809}a-\frac{85532845077}{1930820643809}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
Rank: | $5$ | 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: | $\frac{7733287187}{1930820643809}a^{7}+\frac{10011865399}{1930820643809}a^{6}-\frac{233738309091}{1930820643809}a^{5}-\frac{888534037993}{1930820643809}a^{4}+\frac{1176367312788}{1930820643809}a^{3}+\frac{41983920894985}{1930820643809}a^{2}+\frac{151096689472588}{1930820643809}a+\frac{105024327262035}{1930820643809}$, $\frac{41331968545}{1930820643809}a^{7}-\frac{37338574175}{1930820643809}a^{6}-\frac{1119700820904}{1930820643809}a^{5}-\frac{2492590342713}{1930820643809}a^{4}+\frac{11003309649501}{1930820643809}a^{3}+\frac{200462830619992}{1930820643809}a^{2}+\frac{380971573690072}{1930820643809}a-\frac{93302462659005}{1930820643809}$, $\frac{25853611557}{1930820643809}a^{7}-\frac{64148087102}{1930820643809}a^{6}-\frac{566644110619}{1930820643809}a^{5}-\frac{747077248130}{1930820643809}a^{4}+\frac{7440866943370}{1930820643809}a^{3}+\frac{114034782107940}{1930820643809}a^{2}+\frac{69104913779814}{1930820643809}a-\frac{20511757778469}{1930820643809}$, $\frac{17187234633}{1930820643809}a^{7}-\frac{62698014876}{1930820643809}a^{6}-\frac{189535621374}{1930820643809}a^{5}-\frac{1054909142441}{1930820643809}a^{4}+\frac{7670995332522}{1930820643809}a^{3}+\frac{51609451566864}{1930820643809}a^{2}+\frac{77362827632050}{1930820643809}a+\frac{34014500019025}{1930820643809}$, $\frac{98256362688}{1930820643809}a^{7}-\frac{234306333623}{1930820643809}a^{6}-\frac{2177790381494}{1930820643809}a^{5}-\frac{3223928000637}{1930820643809}a^{4}+\frac{28433146999851}{1930820643809}a^{3}+\frac{437286169808904}{1930820643809}a^{2}+\frac{277356308093502}{1930820643809}a-\frac{82089564237469}{1930820643809}$ (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: | \( 144495.61499 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 144495.61499 \cdot 2}{2\cdot\sqrt{14513641568075776}}\cr\approx \mathstrut & 0.75761085817 \end{aligned}\] (assuming GRH)
Galois group
$C_2^3:\GL(3,2)$ (as 8T48):
A non-solvable group of order 1344 |
The 11 conjugacy class representatives for $C_2^3:\GL(3,2)$ |
Character table for $C_2^3:\GL(3,2)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 8 sibling: | 8.0.907102598004736.7 |
Degree 14 siblings: | deg 14, deg 14 |
Degree 28 siblings: | deg 28, deg 28, deg 28 |
Degree 42 siblings: | deg 42, deg 42, deg 42, some data not computed |
Minimal sibling: | 8.0.907102598004736.7 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.20.58 | $x^{8} + 4 x^{6} + 4 x^{5} + 2 x^{4} + 14$ | $8$ | $1$ | $20$ | $Q_8:C_2$ | $[2, 3, 3]^{2}$ |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.7.12.1 | $x^{7} + 42 x^{6} + 7$ | $7$ | $1$ | $12$ | $C_7$ | $[2]$ |