Normalized defining polynomial
\( x^{8} - 3x^{7} + 57x^{6} + 135x^{5} + 306x^{4} + 5365x^{3} + 23383x^{2} + 40951x + 75421 \)
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: | \(805005849390625\) \(\medspace = 5^{6}\cdot 61^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(72.98\) | 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: | $5^{3/4}61^{3/4}\approx 72.98353242310733$ | ||
Ramified primes: | \(5\), \(61\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | 8.0.805005849390625.1$^{8}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{76}a^{6}-\frac{9}{76}a^{5}-\frac{13}{38}a^{4}+\frac{1}{19}a^{3}-\frac{8}{19}a^{2}-\frac{7}{76}a-\frac{7}{76}$, $\frac{1}{2776873748632}a^{7}-\frac{4218217855}{1388436874316}a^{6}+\frac{455583057451}{2776873748632}a^{5}-\frac{353240259185}{1388436874316}a^{4}-\frac{84896876892}{347109218579}a^{3}+\frac{151184483669}{2776873748632}a^{2}-\frac{153350885244}{347109218579}a+\frac{827222192383}{2776873748632}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{6}\times C_{6}$, which has order $72$
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: | $\frac{613817}{18636736568}a^{7}-\frac{1919613}{9318368284}a^{6}+\frac{39330795}{18636736568}a^{5}-\frac{15368441}{9318368284}a^{4}-\frac{16892172}{2329592071}a^{3}+\frac{2971780125}{18636736568}a^{2}+\frac{704202095}{4659184142}a+\frac{3118694423}{18636736568}$, $\frac{43798701}{1388436874316}a^{7}-\frac{289710369}{694218437158}a^{6}-\frac{1738960249}{1388436874316}a^{5}-\frac{12391936053}{694218437158}a^{4}-\frac{48214800492}{347109218579}a^{3}-\frac{622714977135}{1388436874316}a^{2}-\frac{270482398305}{347109218579}a-\frac{1528242901129}{1388436874316}$, $\frac{5825}{10848392}a^{7}-\frac{16725}{5424196}a^{6}+\frac{417635}{10848392}a^{5}-\frac{35525}{5424196}a^{4}+\frac{63375}{1356049}a^{3}+\frac{36354525}{10848392}a^{2}+\frac{9815225}{2712098}a+\frac{163315359}{10848392}$ | 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: | \( 194.255649708 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 194.255649708 \cdot 72}{2\cdot\sqrt{805005849390625}}\cr\approx \mathstrut & 0.384145822580 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $Q_8$ |
Character table for $Q_8$ |
Intermediate fields
\(\Q(\sqrt{305}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{5}, \sqrt{61})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.1.0.1}{1} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.1.0.1}{1} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(61\) | 61.4.3.1 | $x^{4} + 183$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
61.4.3.1 | $x^{4} + 183$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.61.2t1.a.a | $1$ | $ 61 $ | \(\Q(\sqrt{61}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.305.2t1.a.a | $1$ | $ 5 \cdot 61 $ | \(\Q(\sqrt{305}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
*2 | 2.93025.8t5.a.a | $2$ | $ 5^{2} \cdot 61^{2}$ | 8.0.805005849390625.1 | $Q_8$ (as 8T5) | $-1$ | $-2$ |