Normalized defining polynomial
\( x^{16} - 4x^{14} + 4x^{12} - 4x^{10} + 10x^{8} + 4x^{6} + 4x^{4} + 4x^{2} + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1584616432828678144\) \(\medspace = 2^{38}\cdot 7^{8}\) | 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: | \(13.72\) | 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: | $2^{19/8}7^{2/3}\approx 18.982129549272347$ | ||
Ramified primes: | \(2\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{14}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{14}-\frac{1}{8}a^{13}-\frac{1}{8}a^{12}-\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{3}{8}a^{7}+\frac{3}{8}a^{6}-\frac{1}{8}a^{5}+\frac{3}{8}a^{4}-\frac{1}{8}a^{3}-\frac{1}{8}a^{2}+\frac{3}{8}a-\frac{1}{8}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -a^{14} + \frac{9}{2} a^{12} - 6 a^{10} + 6 a^{8} - 12 a^{6} + \frac{3}{2} a^{4} - 3 a^{2} - 2 \) (order $4$) | 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: | $a$, $\frac{3}{2}a^{15}-\frac{13}{2}a^{13}+\frac{33}{4}a^{11}-\frac{37}{4}a^{9}+19a^{7}-a^{5}+\frac{27}{4}a^{3}+\frac{17}{4}a$, $\frac{3}{8}a^{15}-\frac{3}{8}a^{14}-\frac{13}{8}a^{13}+\frac{11}{8}a^{12}+\frac{17}{8}a^{11}-\frac{7}{8}a^{10}-\frac{21}{8}a^{9}+\frac{5}{8}a^{8}+\frac{43}{8}a^{7}-\frac{27}{8}a^{6}-\frac{9}{8}a^{5}-\frac{17}{8}a^{4}+\frac{21}{8}a^{3}-\frac{11}{8}a^{2}+\frac{11}{8}a-\frac{11}{8}$, $\frac{1}{8}a^{15}+\frac{3}{8}a^{14}-\frac{3}{8}a^{13}-\frac{15}{8}a^{12}-\frac{1}{8}a^{11}+\frac{25}{8}a^{10}+\frac{3}{8}a^{9}-\frac{27}{8}a^{8}+\frac{5}{8}a^{7}+\frac{43}{8}a^{6}+\frac{17}{8}a^{5}-\frac{19}{8}a^{4}-\frac{1}{8}a^{3}+\frac{5}{8}a^{2}-\frac{5}{8}a+\frac{5}{8}$, $\frac{1}{8}a^{15}+\frac{7}{8}a^{14}-\frac{7}{8}a^{13}-\frac{29}{8}a^{12}+\frac{19}{8}a^{11}+\frac{31}{8}a^{10}-\frac{27}{8}a^{9}-\frac{29}{8}a^{8}+\frac{29}{8}a^{7}+\frac{75}{8}a^{6}-\frac{31}{8}a^{5}+\frac{15}{8}a^{4}+\frac{19}{8}a^{3}+\frac{15}{8}a^{2}+\frac{9}{8}a+\frac{11}{8}$, $\frac{9}{8}a^{15}+\frac{1}{8}a^{14}-\frac{39}{8}a^{13}-\frac{5}{8}a^{12}+\frac{49}{8}a^{11}+\frac{7}{8}a^{10}-\frac{53}{8}a^{9}-\frac{5}{8}a^{8}+\frac{109}{8}a^{7}+\frac{13}{8}a^{6}+\frac{1}{8}a^{5}-\frac{5}{8}a^{4}+\frac{33}{8}a^{3}-\frac{9}{8}a^{2}+\frac{23}{8}a-\frac{1}{8}$, $\frac{7}{8}a^{15}+\frac{1}{8}a^{14}-\frac{27}{8}a^{13}-\frac{7}{8}a^{12}+\frac{23}{8}a^{11}+\frac{19}{8}a^{10}-\frac{21}{8}a^{9}-\frac{27}{8}a^{8}+\frac{67}{8}a^{7}+\frac{29}{8}a^{6}+\frac{37}{8}a^{5}-\frac{31}{8}a^{4}+\frac{23}{8}a^{3}+\frac{19}{8}a^{2}+\frac{15}{8}a+\frac{9}{8}$ | 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: | \( 1412.0195181763331 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1412.0195181763331 \cdot 1}{4\cdot\sqrt{1584616432828678144}}\cr\approx \mathstrut & 0.681173239254618 \end{aligned}\]
Galois group
$\SL(2,3):C_2$ (as 16T60):
A solvable group of order 48 |
The 14 conjugacy class representatives for $\SL(2,3):C_2$ |
Character table for $\SL(2,3):C_2$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), 4.0.3136.1, 8.0.39337984.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 24 sibling: | deg 24 |
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 | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.3.0.1}{3} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $16$ | $16$ | $1$ | $38$ | |||
\(7\) | 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
7.12.8.1 | $x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
1.28.6t1.a.a | $1$ | $ 2^{2} \cdot 7 $ | 6.0.153664.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
1.28.6t1.a.b | $1$ | $ 2^{2} \cdot 7 $ | 6.0.153664.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
2.3136.24t21.a.a | $2$ | $ 2^{6} \cdot 7^{2}$ | 16.0.1584616432828678144.8 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ | |
2.3136.24t21.a.b | $2$ | $ 2^{6} \cdot 7^{2}$ | 16.0.1584616432828678144.8 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ | |
* | 2.448.16t60.a.a | $2$ | $ 2^{6} \cdot 7 $ | 16.0.1584616432828678144.8 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ |
* | 2.448.16t60.a.b | $2$ | $ 2^{6} \cdot 7 $ | 16.0.1584616432828678144.8 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ |
* | 2.448.16t60.a.c | $2$ | $ 2^{6} \cdot 7 $ | 16.0.1584616432828678144.8 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ |
* | 2.448.16t60.a.d | $2$ | $ 2^{6} \cdot 7 $ | 16.0.1584616432828678144.8 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ |
* | 3.3136.6t6.a.a | $3$ | $ 2^{6} \cdot 7^{2}$ | 6.4.153664.1 | $A_4\times C_2$ (as 6T6) | $1$ | $1$ |
* | 3.3136.4t4.a.a | $3$ | $ 2^{6} \cdot 7^{2}$ | 4.0.3136.1 | $A_4$ (as 4T4) | $1$ | $-1$ |