Normalized defining polynomial
\( x^{8} - 36x^{6} + 358x^{4} - 1332x^{2} + 1369 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(57420021760000\) \(\medspace = 2^{26}\cdot 5^{4}\cdot 37^{2}\) | 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: | \(52.47\) | 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))
| |
Galois root discriminant: | $2^{55/16}5^{1/2}37^{1/2}\approx 147.35893506362828$ | ||
Ramified primes: | \(2\), \(5\), \(37\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{6}a^{4}-\frac{1}{6}$, $\frac{1}{6}a^{5}-\frac{1}{6}a$, $\frac{1}{444}a^{6}+\frac{1}{444}a^{4}-\frac{49}{444}a^{2}-\frac{1}{12}$, $\frac{1}{444}a^{7}+\frac{1}{444}a^{5}-\frac{49}{444}a^{3}-\frac{1}{12}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
$C_{2}\times C_{2}$, which has order $4$
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: | $\frac{1}{74}a^{6}-\frac{18}{37}a^{4}+\frac{321}{74}a^{2}-6$, $\frac{7}{444}a^{6}+\frac{215}{444}a^{4}-\frac{1433}{444}a^{2}+\frac{97}{12}$, $\frac{1}{222}a^{7}+\frac{1}{148}a^{6}+\frac{6}{37}a^{5}-\frac{71}{444}a^{4}-\frac{142}{111}a^{3}+\frac{173}{148}a^{2}+\frac{5}{2}a-\frac{31}{12}$, $\frac{1}{222}a^{7}-\frac{6}{37}a^{5}+\frac{395}{222}a^{3}-a^{2}-6a+6$, $\frac{5}{37}a^{7}-\frac{27}{74}a^{6}+\frac{895}{222}a^{5}+\frac{412}{37}a^{4}-\frac{1693}{74}a^{3}-\frac{2539}{37}a^{2}+\frac{67}{3}a+\frac{175}{2}$ | 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: | \( 5181.033277591456 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 5181.033277591456 \cdot 4}{2\cdot\sqrt{57420021760000}}\cr\approx \mathstrut & 0.863762649302484 \end{aligned}\]
Galois group
$C_2\wr C_2^2$ (as 8T29):
A solvable group of order 64 |
The 16 conjugacy class representatives for $(((C_4 \times C_2): C_2):C_2):C_2$ |
Character table for $(((C_4 \times C_2): C_2):C_2):C_2$ |
Intermediate fields
\(\Q(\sqrt{10}) \), 4.2.25600.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 32 siblings: | 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 | ${\href{/padicField/3.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.26.35 | $x^{8} + 4 x^{6} + 8 x^{3} + 10$ | $8$ | $1$ | $26$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $[2, 2, 3, 7/2, 4]^{2}$ |
\(5\) | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(37\) | $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
37.2.1.1 | $x^{2} + 37$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |