Normalized defining polynomial
\( x^{8} - 2x^{7} + 14x^{6} - 28x^{5} - 77x^{4} - 420x^{3} + 630x^{2} + 4650x + 4721 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1043797559316736\) \(\medspace = 2^{8}\cdot 7^{8}\cdot 29^{4}\) | 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: | \(75.39\) | 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: | $2\cdot 7^{26/21}29^{1/2}\approx 119.82318344449011$ | ||
Ramified primes: | \(2\), \(7\), \(29\) | 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)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{6}-\frac{3}{16}a^{5}+\frac{7}{16}a^{3}-\frac{1}{4}a^{2}-\frac{7}{16}a+\frac{1}{16}$, $\frac{1}{125300368}a^{7}+\frac{675327}{31325092}a^{6}-\frac{8061357}{125300368}a^{5}-\frac{14890969}{125300368}a^{4}+\frac{53669573}{125300368}a^{3}+\frac{13441469}{125300368}a^{2}+\frac{332611}{15662546}a+\frac{4263791}{125300368}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{4}$, which has order $8$ (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)
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Fundamental units: | $\frac{19317495}{125300368}a^{7}-\frac{79642547}{125300368}a^{6}+\frac{215511429}{62650184}a^{5}-\frac{1407776959}{125300368}a^{4}+\frac{603696105}{62650184}a^{3}-\frac{9861647161}{125300368}a^{2}+\frac{30239141801}{125300368}a+\frac{13819766225}{62650184}$, $\frac{1227191}{15662546}a^{7}+\frac{38619777}{125300368}a^{6}+\frac{286028869}{125300368}a^{5}+\frac{134094691}{15662546}a^{4}+\frac{3078070783}{125300368}a^{3}+\frac{1034947741}{31325092}a^{2}+\frac{478954761}{125300368}a-\frac{2268436855}{125300368}$, $\frac{16168861}{62650184}a^{7}-\frac{120875979}{62650184}a^{6}+\frac{133890813}{15662546}a^{5}-\frac{1591793269}{62650184}a^{4}+\frac{9684668}{7831273}a^{3}+\frac{9810955341}{62650184}a^{2}-\frac{4547681175}{62650184}a-\frac{5274910847}{15662546}$ (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: | \( 6234.12181252 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 6234.12181252 \cdot 8}{2\cdot\sqrt{1043797559316736}}\cr\approx \mathstrut & 1.20294732358 \end{aligned}\] (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: | 7.3.77571162256.4, 7.3.77571162256.3 |
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, some data not computed |
Minimal sibling: | 7.3.77571162256.4 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | R | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ |
2.4.4.2 | $x^{4} + 4 x^{3} + 4 x^{2} + 12$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.7.8.1 | $x^{7} + 14 x^{2} + 7$ | $7$ | $1$ | $8$ | $C_7:C_3$ | $[4/3]_{3}$ | |
\(29\) | 29.4.2.2 | $x^{4} - 696 x^{2} + 1682$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
29.4.2.2 | $x^{4} - 696 x^{2} + 1682$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |