Normalized defining polynomial
\( x^{8} - x^{7} - 11x^{6} + 10x^{5} + 41x^{4} - 38x^{3} - 64x^{2} + 58x + 53 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(4546939761\) \(\medspace = 3^{2}\cdot 7^{2}\cdot 13^{4}\cdot 19^{2}\) | 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: | \(16.11\) | 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))
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Galois root discriminant: | $3^{1/2}7^{1/2}13^{1/2}19^{1/2}\approx 72.02083032012335$ | ||
Ramified primes: | \(3\), \(7\), \(13\), \(19\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{21}a^{7}+\frac{1}{7}a^{5}+\frac{2}{7}a^{4}-\frac{2}{21}a^{3}-\frac{5}{21}a^{2}+\frac{1}{21}a-\frac{4}{21}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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)
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Fundamental units: | $\frac{2}{21}a^{7}-\frac{1}{3}a^{6}-\frac{8}{21}a^{5}+\frac{47}{21}a^{4}-\frac{6}{7}a^{3}-\frac{29}{7}a^{2}+\frac{24}{7}a+\frac{41}{21}$, $\frac{4}{21}a^{7}-\frac{1}{3}a^{6}-\frac{10}{7}a^{5}+\frac{59}{21}a^{4}+\frac{55}{21}a^{3}-\frac{160}{21}a^{2}-\frac{1}{7}a+\frac{110}{21}$, $\frac{1}{21}a^{7}-\frac{1}{3}a^{6}+\frac{10}{21}a^{5}+\frac{20}{21}a^{4}-\frac{58}{21}a^{3}+\frac{23}{21}a^{2}+\frac{50}{21}a-\frac{13}{7}$, $\frac{1}{7}a^{7}-\frac{1}{3}a^{6}-\frac{19}{21}a^{5}+\frac{53}{21}a^{4}+\frac{8}{21}a^{3}-\frac{113}{21}a^{2}+\frac{29}{7}a+\frac{44}{21}$, $\frac{4}{21}a^{7}-\frac{2}{3}a^{6}-\frac{10}{7}a^{5}+\frac{115}{21}a^{4}+\frac{55}{21}a^{3}-\frac{100}{7}a^{2}+\frac{32}{21}a+\frac{320}{21}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 64.5079170739 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 64.5079170739 \cdot 1}{2\cdot\sqrt{4546939761}}\cr\approx \mathstrut & 0.302136464123 \end{aligned}\]
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: | 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 | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(19\) | 19.4.2.2 | $x^{4} - 2888 x^{3} - 767106 x^{2} - 76532 x + 722$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.39.2t1.a.a | $1$ | $ 3 \cdot 13 $ | \(\Q(\sqrt{-39}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.399.2t1.a.a | $1$ | $ 3 \cdot 7 \cdot 19 $ | \(\Q(\sqrt{-399}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.1729.2t1.a.a | $1$ | $ 7 \cdot 13 \cdot 19 $ | \(\Q(\sqrt{1729}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.13.2t1.a.a | $1$ | $ 13 $ | \(\Q(\sqrt{13}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.133.2t1.a.a | $1$ | $ 7 \cdot 19 $ | \(\Q(\sqrt{133}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.5187.2t1.a.a | $1$ | $ 3 \cdot 7 \cdot 13 \cdot 19 $ | \(\Q(\sqrt{-5187}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.39.4t3.b.a | $2$ | $ 3 \cdot 13 $ | 4.2.507.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
2.399.4t3.f.a | $2$ | $ 3 \cdot 7 \cdot 19 $ | 4.2.53067.2 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.5187.4t3.j.a | $2$ | $ 3 \cdot 7 \cdot 13 \cdot 19 $ | 4.2.8968323.2 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.5187.4t3.i.a | $2$ | $ 3 \cdot 7 \cdot 13 \cdot 19 $ | 4.2.8968323.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.67431.4t3.c.a | $2$ | $ 3 \cdot 7 \cdot 13^{2} \cdot 19 $ | 4.2.8968323.6 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
2.689871.4t3.b.a | $2$ | $ 3 \cdot 7^{2} \cdot 13 \cdot 19^{2}$ | 4.2.8968323.5 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.80714907.8t29.e.a | $4$ | $ 3^{3} \cdot 7^{2} \cdot 13^{2} \cdot 19^{2}$ | 8.4.4546939761.1 | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $-2$ | |
* | 4.8968323.8t29.e.a | $4$ | $ 3 \cdot 7^{2} \cdot 13^{2} \cdot 19^{2}$ | 8.4.4546939761.1 | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $2$ |