Normalized defining polynomial
\( x^{8} - x^{7} - 13x^{6} - 13x^{5} - 20x^{4} + 358x^{3} + 227x^{2} - 539x + 991 \)
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: | \(211922578125\) \(\medspace = 3^{6}\cdot 5^{7}\cdot 61^{2}\) | 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: | \(26.05\) | 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: | $3^{3/4}5^{7/8}61^{1/2}\approx 72.79551323875552$ | ||
Ramified primes: | \(3\), \(5\), \(61\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{29}a^{6}-\frac{10}{29}a^{5}+\frac{1}{29}a^{4}+\frac{13}{29}a^{3}-\frac{10}{29}a^{2}+\frac{11}{29}a-\frac{11}{29}$, $\frac{1}{94140931}a^{7}+\frac{149320}{94140931}a^{6}+\frac{24445}{1595609}a^{5}-\frac{9521374}{94140931}a^{4}-\frac{8515917}{94140931}a^{3}-\frac{20963309}{94140931}a^{2}+\frac{43803631}{94140931}a-\frac{25499654}{94140931}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $5$ | 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: | $\frac{17}{55021}a^{7}-\frac{3338}{1595609}a^{6}-\frac{4668}{1595609}a^{5}+\frac{24212}{1595609}a^{4}+\frac{30324}{1595609}a^{3}+\frac{453387}{1595609}a^{2}-\frac{607438}{1595609}a-\frac{1892014}{1595609}$, $\frac{15459}{94140931}a^{7}+\frac{261951}{94140931}a^{6}+\frac{11027}{1595609}a^{5}-\frac{3198167}{94140931}a^{4}-\frac{12569453}{94140931}a^{3}-\frac{22478134}{94140931}a^{2}+\frac{33831097}{94140931}a+\frac{222133553}{94140931}$, $\frac{149768}{94140931}a^{7}+\frac{17289}{94140931}a^{6}-\frac{18580}{1595609}a^{5}-\frac{2258268}{94140931}a^{4}-\frac{6755263}{94140931}a^{3}+\frac{15880123}{94140931}a^{2}-\frac{23343467}{94140931}a-\frac{104680848}{94140931}$, $\frac{1153035}{94140931}a^{7}+\frac{3654596}{94140931}a^{6}-\frac{52242}{1595609}a^{5}-\frac{28288468}{94140931}a^{4}-\frac{134390191}{94140931}a^{3}-\frac{60002136}{94140931}a^{2}+\frac{4271936}{3246239}a-\frac{264804977}{94140931}$, $\frac{1817789}{94140931}a^{7}+\frac{4471973}{94140931}a^{6}-\frac{152894}{1595609}a^{5}-\frac{58235170}{94140931}a^{4}-\frac{235324239}{94140931}a^{3}-\frac{123397419}{94140931}a^{2}+\frac{109350044}{94140931}a-\frac{511955831}{94140931}$ | 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: | \( 218.828962683 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 218.828962683 \cdot 2}{2\cdot\sqrt{211922578125}}\cr\approx \mathstrut & 0.300258912078 \end{aligned}\]
Galois group
$\OD_{16}$ (as 8T7):
A solvable group of order 16 |
The 10 conjugacy class representatives for $C_8:C_2$ |
Character table for $C_8:C_2$ |
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
Galois closure: | deg 16 |
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 | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.1.0.1}{1} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.6.3 | $x^{8} - 6 x^{4} + 18$ | $4$ | $2$ | $6$ | $C_8:C_2$ | $[\ ]_{4}^{4}$ |
\(5\) | 5.8.7.1 | $x^{8} + 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
\(61\) | 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
61.2.1.2 | $x^{2} + 122$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
61.2.1.2 | $x^{2} + 122$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.915.2t1.a.a | $1$ | $ 3 \cdot 5 \cdot 61 $ | \(\Q(\sqrt{-915}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.183.2t1.a.a | $1$ | $ 3 \cdot 61 $ | \(\Q(\sqrt{-183}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.305.4t1.a.a | $1$ | $ 5 \cdot 61 $ | 4.0.465125.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
* | 1.15.4t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
* | 1.15.4t1.a.b | $1$ | $ 3 \cdot 5 $ | \(\Q(\zeta_{15})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
1.305.4t1.a.b | $1$ | $ 5 \cdot 61 $ | 4.0.465125.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
* | 2.13725.8t7.b.a | $2$ | $ 3^{2} \cdot 5^{2} \cdot 61 $ | 8.4.211922578125.2 | $C_8:C_2$ (as 8T7) | $0$ | $0$ |
* | 2.13725.8t7.b.b | $2$ | $ 3^{2} \cdot 5^{2} \cdot 61 $ | 8.4.211922578125.2 | $C_8:C_2$ (as 8T7) | $0$ | $0$ |