Normalized defining polynomial
\( x^{8} - 4x^{7} - 28x^{6} - 28x^{5} + 70x^{4} + 588x^{3} + 1764x^{2} + 1940x + 661 \)
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: | \(3628410392018944\) \(\medspace = 2^{18}\cdot 7^{12}\) | 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: | \(88.10\) | 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: | $2^{11/4}7^{12/7}\approx 189.04795562662562$ | ||
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4}a^{4}+\frac{1}{4}$, $\frac{1}{8}a^{5}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}+\frac{1}{8}a+\frac{3}{8}$, $\frac{1}{8}a^{6}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}+\frac{1}{8}a^{2}-\frac{1}{2}a-\frac{1}{8}$, $\frac{1}{57320}a^{7}+\frac{1319}{57320}a^{6}-\frac{3251}{57320}a^{5}-\frac{2101}{57320}a^{4}+\frac{14797}{57320}a^{3}-\frac{26421}{57320}a^{2}-\frac{16679}{57320}a+\frac{3823}{57320}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (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: | $\frac{18861}{57320}a^{7}-\frac{85201}{57320}a^{6}-\frac{486261}{57320}a^{5}-\frac{262451}{57320}a^{4}+\frac{1485457}{57320}a^{3}+\frac{10302539}{57320}a^{2}+\frac{27804071}{57320}a+\frac{21248433}{57320}$, $\frac{2249}{11464}a^{7}-\frac{14209}{11464}a^{6}-\frac{15213}{5732}a^{5}+\frac{2729}{2866}a^{4}+\frac{130297}{11464}a^{3}+\frac{1017379}{11464}a^{2}+\frac{791301}{5732}a+\frac{156537}{2866}$, $\frac{279}{11464}a^{7}-\frac{1713}{11464}a^{6}-\frac{709}{1433}a^{5}+\frac{5691}{5732}a^{4}+\frac{29983}{11464}a^{3}+\frac{42883}{11464}a^{2}+\frac{36419}{2866}a+\frac{55403}{5732}$, $\frac{2047}{28660}a^{7}-\frac{8377}{28660}a^{6}-\frac{118829}{57320}a^{5}-\frac{67979}{57320}a^{4}+\frac{182129}{28660}a^{3}+\frac{1187023}{28660}a^{2}+\frac{6927179}{57320}a+\frac{5355257}{57320}$, $\frac{343869}{57320}a^{7}-\frac{811037}{28660}a^{6}-\frac{4226847}{28660}a^{5}-\frac{3632979}{57320}a^{4}+\frac{26472353}{57320}a^{3}+\frac{91665653}{28660}a^{2}+\frac{237934317}{28660}a+\frac{331128957}{57320}$ (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: | \( 146393.854072 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 146393.854072 \cdot 1}{2\cdot\sqrt{3628410392018944}}\cr\approx \mathstrut & 0.767563592998 \end{aligned}\] (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: | 8.0.14513641568075776.7 |
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: | 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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.18.1 | $x^{8} + 8 x^{7} + 40 x^{6} + 112 x^{5} + 232 x^{4} + 240 x^{3} + 256 x^{2} + 96 x + 84$ | $4$ | $2$ | $18$ | $D_4\times C_2$ | $[2, 3, 7/2]^{2}$ |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.7.12.1 | $x^{7} + 42 x^{6} + 7$ | $7$ | $1$ | $12$ | $C_7$ | $[2]$ |