Normalized defining polynomial
\( x^{8} - x^{7} + 10x^{6} - 9x^{5} + 79x^{4} - 59x^{3} + 180x^{2} + 99x + 121 \)
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: | \(3038765625\) \(\medspace = 3^{4}\cdot 5^{6}\cdot 7^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(15.32\) | 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: | $3^{1/2}5^{3/4}7^{1/2}\approx 15.322765339111537$ | ||
Ramified primes: | \(3\), \(5\), \(7\) | 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 and abelian over $\Q$. | |||
Conductor: | \(105=3\cdot 5\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{105}(64,·)$, $\chi_{105}(1,·)$, $\chi_{105}(97,·)$, $\chi_{105}(71,·)$, $\chi_{105}(13,·)$, $\chi_{105}(83,·)$, $\chi_{105}(29,·)$, $\chi_{105}(62,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), 4.0.55125.1$^{2}$, 8.0.3038765625.1$^{4}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{1387969}a^{7}+\frac{1803}{1387969}a^{6}+\frac{476684}{1387969}a^{5}-\frac{602853}{1387969}a^{4}+\frac{620963}{1387969}a^{3}+\frac{126210}{1387969}a^{2}+\frac{56104}{1387969}a-\frac{10002}{126179}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{7690}{1387969} a^{7} - \frac{14620}{1387969} a^{6} + \frac{73831}{1387969} a^{5} - \frac{123110}{1387969} a^{4} + \frac{592110}{1387969} a^{3} - \frac{1023400}{1387969} a^{2} + \frac{1169370}{1387969} a + \frac{53810}{126179} \) (order $6$) | 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: | $\frac{26162}{1387969}a^{7}-\frac{20860}{1387969}a^{6}+\frac{105343}{1387969}a^{5}-\frac{348439}{1387969}a^{4}+\frac{844830}{1387969}a^{3}-\frac{1460200}{1387969}a^{2}-\frac{2066323}{1387969}a-\frac{103257}{126179}$, $\frac{62303}{1387969}a^{7}-\frac{93180}{1387969}a^{6}+\frac{470559}{1387969}a^{5}-\frac{1109319}{1387969}a^{4}+\frac{3773790}{1387969}a^{3}-\frac{6522600}{1387969}a^{2}+\frac{4705477}{1387969}a-\frac{461241}{126179}$, $\frac{33852}{1387969}a^{7}-\frac{35480}{1387969}a^{6}+\frac{179174}{1387969}a^{5}-\frac{471549}{1387969}a^{4}+\frac{1436940}{1387969}a^{3}-\frac{2483600}{1387969}a^{2}-\frac{2284922}{1387969}a-\frac{175626}{126179}$ | 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: | \( 26.9289188006 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 26.9289188006 \cdot 2}{6\cdot\sqrt{3038765625}}\cr\approx \mathstrut & 0.253786509728 \end{aligned}\]
Galois group
$C_2\times C_4$ (as 8T2):
An abelian group of order 8 |
The 8 conjugacy class representatives for $C_4\times C_2$ |
Character table for $C_4\times C_2$ |
Intermediate fields
\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 4.0.55125.1, 4.4.6125.1 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | R | R | R | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.1.0.1}{1} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(5\) | 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
\(7\) | 7.4.2.2 | $x^{4} - 42 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
7.4.2.2 | $x^{4} - 42 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |