Normalized defining polynomial
\( x^{8} - 3x^{7} + 33x^{6} - 56x^{5} + 350x^{4} - 306x^{3} + 1678x^{2} - 633x + 3641 \)
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)
| |
Discriminant: | \(14512578125\) \(\medspace = 5^{7}\cdot 431^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.63\) | 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: | $5^{7/8}431^{1/2}\approx 84.88625793449107$ | ||
Ramified primes: | \(5\), \(431\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{45249781}a^{7}+\frac{4219770}{45249781}a^{6}+\frac{3942028}{45249781}a^{5}+\frac{10327054}{45249781}a^{4}+\frac{22047042}{45249781}a^{3}+\frac{8074941}{45249781}a^{2}+\frac{20673422}{45249781}a+\frac{4683111}{45249781}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{216503}{45249781} a^{7} + \frac{214080}{45249781} a^{6} - \frac{4768643}{45249781} a^{5} - \frac{1243171}{45249781} a^{4} - \frac{32335560}{45249781} a^{3} - \frac{23662388}{45249781} a^{2} - \frac{111544994}{45249781} a - \frac{40987747}{45249781} \) (order $10$) | 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: | $\frac{233274}{45249781}a^{7}-\frac{1108894}{45249781}a^{6}+\frac{6590190}{45249781}a^{5}-\frac{19895863}{45249781}a^{4}+\frac{47316391}{45249781}a^{3}-\frac{119845977}{45249781}a^{2}+\frac{121683334}{45249781}a-\frac{379425517}{45249781}$, $\frac{468061}{45249781}a^{7}+\frac{2075101}{45249781}a^{6}+\frac{4497652}{45249781}a^{5}+\frac{64366093}{45249781}a^{4}+\frac{12219169}{45249781}a^{3}+\frac{484249405}{45249781}a^{2}+\frac{164155921}{45249781}a+\frac{1575468904}{45249781}$, $\frac{1872706}{45249781}a^{7}+\frac{12093561}{45249781}a^{6}+\frac{29216304}{45249781}a^{5}+\frac{322586096}{45249781}a^{4}+\frac{279658260}{45249781}a^{3}+\frac{2409636130}{45249781}a^{2}+\frac{1429287134}{45249781}a+\frac{6655481658}{45249781}$ | 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: | \( 76.8982230396 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 76.8982230396 \cdot 3}{10\cdot\sqrt{14512578125}}\cr\approx \mathstrut & 0.298459046420 \end{aligned}\]
Galois group
$\OD_{16}:C_2$ (as 8T16):
A solvable group of order 32 |
The 11 conjugacy class representatives for $(C_8:C_2):C_2$ |
Character table for $(C_8:C_2):C_2$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 8 sibling: | data not computed |
Degree 16 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 | ${\href{/padicField/2.8.0.1}{8} }$ | ${\href{/padicField/3.8.0.1}{8} }$ | R | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.8.7.2 | $x^{8} + 5$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
\(431\) | $\Q_{431}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{431}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.2155.2t1.a.a | $1$ | $ 5 \cdot 431 $ | \(\Q(\sqrt{-2155}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.431.2t1.a.a | $1$ | $ 431 $ | \(\Q(\sqrt{-431}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.2155.4t1.a.a | $1$ | $ 5 \cdot 431 $ | 4.4.23220125.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
* | 1.5.4t1.a.a | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
1.2155.4t1.a.b | $1$ | $ 5 \cdot 431 $ | 4.4.23220125.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
* | 1.5.4t1.a.b | $1$ | $ 5 $ | \(\Q(\zeta_{5})\) | $C_4$ (as 4T1) | $0$ | $-1$ |
2.2155.4t3.a.a | $2$ | $ 5 \cdot 431 $ | 4.0.928805.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.10775.4t3.a.a | $2$ | $ 5^{2} \cdot 431 $ | 4.0.23220125.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 4.116100625.8t16.b.a | $4$ | $ 5^{4} \cdot 431^{2}$ | 8.0.14512578125.1 | $(C_8:C_2):C_2$ (as 8T16) | $1$ | $0$ |