Normalized defining polynomial
\( x^{8} - 3x^{7} + 17x^{6} - 30x^{5} + 106x^{4} - 105x^{3} + 273x^{2} - 99x + 261 \)
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: | \(50651578125\) \(\medspace = 3^{4}\cdot 5^{6}\cdot 31\cdot 1291\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(21.78\) | 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: | $3^{1/2}5^{3/4}31^{1/2}1291^{1/2}\approx 1158.596197090594$ | ||
Ramified primes: | \(3\), \(5\), \(31\), \(1291\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{40021}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | deg 16$^{8}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{165}a^{6}-\frac{9}{55}a^{5}+\frac{56}{165}a^{4}+\frac{18}{55}a^{3}+\frac{16}{165}a^{2}-\frac{3}{11}a+\frac{8}{55}$, $\frac{1}{825}a^{7}-\frac{13}{825}a^{5}-\frac{28}{275}a^{4}-\frac{16}{75}a^{3}+\frac{74}{275}a^{2}-\frac{122}{275}a+\frac{51}{275}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{12}$, which has order $12$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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: | $\frac{4}{165}a^{7}-\frac{2}{33}a^{6}+\frac{53}{165}a^{5}-\frac{71}{165}a^{4}+\frac{241}{165}a^{3}-\frac{97}{165}a^{2}+\frac{102}{55}a+\frac{69}{55}$, $\frac{2}{275}a^{7}-\frac{1}{165}a^{6}+\frac{19}{275}a^{5}+\frac{41}{825}a^{4}+\frac{108}{275}a^{3}+\frac{427}{825}a^{2}+\frac{168}{275}a+\frac{266}{275}$, $\frac{7}{825}a^{7}-\frac{8}{165}a^{6}+\frac{164}{825}a^{5}-\frac{353}{825}a^{4}+\frac{733}{825}a^{3}-\frac{736}{825}a^{2}+\frac{296}{275}a+\frac{37}{275}$ | 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: | \( 9.32364155459 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 9.32364155459 \cdot 12}{2\cdot\sqrt{50651578125}}\cr\approx \mathstrut & 0.387400165690 \end{aligned}\]
Galois group
$C_2\wr C_4$ (as 8T27):
A solvable group of order 64 |
The 13 conjugacy class representatives for $((C_8 : C_2):C_2):C_2$ |
Character table for $((C_8 : C_2):C_2):C_2$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\) |
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 | ${\href{/padicField/2.8.0.1}{8} }$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(5\) | 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(31\) | $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
31.2.1.1 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(1291\) | $\Q_{1291}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1291}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{1291}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{1291}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{1291}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{1291}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |