Normalized defining polynomial
\( x^{8} - 3x^{7} + 17x^{6} - 35x^{5} + 100x^{4} - 89x^{3} + 313x^{2} + 63x + 321 \)
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: | \(27485992521\) \(\medspace = 3^{4}\cdot 13^{4}\cdot 109^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(20.18\) | 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: | $3^{1/2}13^{1/2}109^{1/2}\approx 65.19969325081216$ | ||
Ramified primes: | \(3\), \(13\), \(109\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{280}a^{6}-\frac{43}{280}a^{5}-\frac{5}{28}a^{4}+\frac{9}{20}a^{3}+\frac{13}{28}a^{2}-\frac{11}{280}a+\frac{3}{280}$, $\frac{1}{37520}a^{7}-\frac{23}{18760}a^{6}-\frac{5241}{37520}a^{5}+\frac{2799}{9380}a^{4}-\frac{13}{70}a^{3}-\frac{11881}{37520}a^{2}-\frac{4191}{9380}a+\frac{12591}{37520}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{4}$, which has order $4$
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: | $\frac{211}{37520}a^{7}-\frac{297}{18760}a^{6}+\frac{3133}{37520}a^{5}-\frac{1691}{9380}a^{4}+\frac{29}{70}a^{3}-\frac{9131}{37520}a^{2}+\frac{9881}{9380}a+\frac{20117}{37520}$, $\frac{117}{18760}a^{7}-\frac{491}{18760}a^{6}+\frac{193}{1876}a^{5}-\frac{1969}{9380}a^{4}+\frac{11}{28}a^{3}-\frac{3847}{18760}a^{2}+\frac{10891}{18760}a+\frac{289}{938}$, $\frac{625}{3752}a^{7}+\frac{9077}{18760}a^{6}+\frac{25037}{9380}a^{5}+\frac{12539}{1876}a^{4}+\frac{1923}{140}a^{3}+\frac{86013}{3752}a^{2}+\frac{335403}{18760}a+\frac{64819}{4690}$ | 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: | \( 34.5772002628 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 34.5772002628 \cdot 4}{2\cdot\sqrt{27485992521}}\cr\approx \mathstrut & 0.650105113965 \end{aligned}\]
Galois group
$C_2\wr C_2^2$ (as 8T29):
A solvable group of order 64 |
The 16 conjugacy class representatives for $(((C_4 \times C_2): C_2):C_2):C_2$ |
Character table for $(((C_4 \times C_2): C_2):C_2):C_2$ |
Intermediate fields
\(\Q(\sqrt{13}) \), 4.2.507.1 |
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: | 8.4.3053999169.1 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | 13.8.4.1 | $x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(109\) | 109.4.0.1 | $x^{4} + 11 x^{2} + 98 x + 6$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
109.4.2.2 | $x^{4} - 11772 x^{2} + 71286$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.4251.2t1.a.a | $1$ | $ 3 \cdot 13 \cdot 109 $ | \(\Q(\sqrt{-4251}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.39.2t1.a.a | $1$ | $ 3 \cdot 13 $ | \(\Q(\sqrt{-39}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.109.2t1.a.a | $1$ | $ 109 $ | \(\Q(\sqrt{109}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.13.2t1.a.a | $1$ | $ 13 $ | \(\Q(\sqrt{13}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.327.2t1.a.a | $1$ | $ 3 \cdot 109 $ | \(\Q(\sqrt{-327}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.1417.2t1.a.a | $1$ | $ 13 \cdot 109 $ | \(\Q(\sqrt{1417}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.55263.4t3.b.a | $2$ | $ 3 \cdot 13^{2} \cdot 109 $ | 4.0.165789.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.327.4t3.c.a | $2$ | $ 3 \cdot 109 $ | 4.0.981.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.4251.4t3.c.a | $2$ | $ 3 \cdot 13 \cdot 109 $ | 4.0.12753.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.463359.4t3.a.a | $2$ | $ 3 \cdot 13 \cdot 109^{2}$ | 4.2.6023667.3 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.4251.4t3.d.a | $2$ | $ 3 \cdot 13 \cdot 109 $ | 4.0.12753.2 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 2.39.4t3.b.a | $2$ | $ 3 \cdot 13 $ | 4.2.507.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
* | 4.54213003.8t29.a.a | $4$ | $ 3^{3} \cdot 13^{2} \cdot 109^{2}$ | 8.0.27485992521.4 | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $-2$ |
4.6023667.8t29.a.a | $4$ | $ 3 \cdot 13^{2} \cdot 109^{2}$ | 8.0.27485992521.4 | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $2$ |