Normalized defining polynomial
\( x^{16} + 86 x^{14} + 2902 x^{12} + 49411 x^{10} + 453495 x^{8} + 2216251 x^{6} + 5273394 x^{4} + \cdots + 1003941 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(92144280422466145326820294656\)
\(\medspace = 2^{16}\cdot 3^{5}\cdot 19^{6}\cdot 103^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(64.61\) | 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: | not computed | ||
Ramified primes: |
\(2\), \(3\), \(19\), \(103\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{309}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | unavailable$^{128}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{3}$, $\frac{1}{33\!\cdots\!67}a^{14}+\frac{30\!\cdots\!72}{33\!\cdots\!67}a^{12}+\frac{55\!\cdots\!71}{33\!\cdots\!67}a^{10}+\frac{96\!\cdots\!90}{33\!\cdots\!67}a^{8}+\frac{76\!\cdots\!62}{11\!\cdots\!89}a^{6}-\frac{11\!\cdots\!34}{33\!\cdots\!67}a^{4}-\frac{77\!\cdots\!68}{37\!\cdots\!63}a^{2}+\frac{76\!\cdots\!76}{19\!\cdots\!77}$, $\frac{1}{33\!\cdots\!67}a^{15}+\frac{30\!\cdots\!72}{33\!\cdots\!67}a^{13}+\frac{55\!\cdots\!71}{33\!\cdots\!67}a^{11}+\frac{96\!\cdots\!90}{33\!\cdots\!67}a^{9}+\frac{76\!\cdots\!62}{11\!\cdots\!89}a^{7}-\frac{11\!\cdots\!34}{33\!\cdots\!67}a^{5}-\frac{77\!\cdots\!68}{37\!\cdots\!63}a^{3}+\frac{76\!\cdots\!76}{19\!\cdots\!77}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{2996}$, which has order $11984$ (assuming GRH)
Unit group
Rank: | $7$ | 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: |
$\frac{8363018672176}{13\!\cdots\!77}a^{14}+\frac{221620768841923}{43\!\cdots\!59}a^{12}+\frac{66\!\cdots\!54}{43\!\cdots\!59}a^{10}+\frac{28\!\cdots\!22}{13\!\cdots\!77}a^{8}+\frac{19\!\cdots\!19}{13\!\cdots\!77}a^{6}+\frac{54\!\cdots\!66}{13\!\cdots\!77}a^{4}+\frac{34\!\cdots\!25}{13\!\cdots\!77}a^{2}-\frac{36\!\cdots\!86}{22\!\cdots\!61}$, $\frac{87\!\cdots\!24}{33\!\cdots\!67}a^{14}+\frac{65\!\cdots\!47}{33\!\cdots\!67}a^{12}+\frac{17\!\cdots\!44}{33\!\cdots\!67}a^{10}+\frac{20\!\cdots\!45}{33\!\cdots\!67}a^{8}+\frac{80\!\cdots\!83}{37\!\cdots\!63}a^{6}-\frac{20\!\cdots\!30}{33\!\cdots\!67}a^{4}-\frac{40\!\cdots\!83}{11\!\cdots\!89}a^{2}-\frac{88\!\cdots\!79}{19\!\cdots\!77}$, $\frac{85\!\cdots\!69}{33\!\cdots\!67}a^{14}+\frac{71\!\cdots\!83}{33\!\cdots\!67}a^{12}+\frac{22\!\cdots\!16}{33\!\cdots\!67}a^{10}+\frac{35\!\cdots\!96}{33\!\cdots\!67}a^{8}+\frac{92\!\cdots\!50}{11\!\cdots\!89}a^{6}+\frac{10\!\cdots\!39}{33\!\cdots\!67}a^{4}+\frac{18\!\cdots\!56}{37\!\cdots\!63}a^{2}+\frac{78\!\cdots\!66}{19\!\cdots\!77}$, $\frac{59\!\cdots\!26}{33\!\cdots\!67}a^{14}+\frac{50\!\cdots\!23}{33\!\cdots\!67}a^{12}+\frac{16\!\cdots\!95}{33\!\cdots\!67}a^{10}+\frac{27\!\cdots\!29}{33\!\cdots\!67}a^{8}+\frac{82\!\cdots\!51}{11\!\cdots\!89}a^{6}+\frac{11\!\cdots\!48}{33\!\cdots\!67}a^{4}+\frac{79\!\cdots\!46}{11\!\cdots\!89}a^{2}+\frac{92\!\cdots\!29}{19\!\cdots\!77}$, $\frac{11\!\cdots\!05}{33\!\cdots\!67}a^{14}+\frac{88\!\cdots\!51}{33\!\cdots\!67}a^{12}+\frac{26\!\cdots\!86}{33\!\cdots\!67}a^{10}+\frac{36\!\cdots\!84}{33\!\cdots\!67}a^{8}+\frac{81\!\cdots\!95}{11\!\cdots\!89}a^{6}+\frac{67\!\cdots\!94}{33\!\cdots\!67}a^{4}+\frac{17\!\cdots\!00}{11\!\cdots\!89}a^{2}+\frac{32\!\cdots\!43}{19\!\cdots\!77}$, $\frac{12\!\cdots\!85}{11\!\cdots\!89}a^{14}+\frac{11\!\cdots\!65}{11\!\cdots\!89}a^{12}+\frac{41\!\cdots\!17}{11\!\cdots\!89}a^{10}+\frac{27\!\cdots\!61}{37\!\cdots\!63}a^{8}+\frac{89\!\cdots\!05}{11\!\cdots\!89}a^{6}+\frac{52\!\cdots\!88}{11\!\cdots\!89}a^{4}+\frac{14\!\cdots\!00}{11\!\cdots\!89}a^{2}+\frac{19\!\cdots\!42}{19\!\cdots\!77}$, $\frac{20\!\cdots\!16}{33\!\cdots\!67}a^{14}+\frac{22\!\cdots\!64}{33\!\cdots\!67}a^{12}+\frac{92\!\cdots\!16}{33\!\cdots\!67}a^{10}+\frac{18\!\cdots\!31}{33\!\cdots\!67}a^{8}+\frac{61\!\cdots\!47}{11\!\cdots\!89}a^{6}+\frac{79\!\cdots\!53}{33\!\cdots\!67}a^{4}+\frac{10\!\cdots\!99}{37\!\cdots\!63}a^{2}+\frac{53\!\cdots\!09}{19\!\cdots\!77}$
| 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: | \( 8552.89250323 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 8552.89250323 \cdot 11984}{2\cdot\sqrt{92144280422466145326820294656}}\cr\approx \mathstrut & 0.410099815370 \end{aligned}\] (assuming GRH)
Galois group
$C_4^3.\GL(2,\mathbb{Z}/4)$ (as 16T1674):
A solvable group of order 6144 |
The 54 conjugacy class representatives for $C_4^3.\GL(2,\mathbb{Z}/4)$ |
Character table for $C_4^3.\GL(2,\mathbb{Z}/4)$ |
Intermediate fields
4.4.1957.1, 8.8.1183423341.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
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 | R | R | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | $16$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $16$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| Deg $16$ | $2$ | $8$ | $16$ | |||
\(3\)
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
3.6.0.1 | $x^{6} + 2 x^{4} + x^{2} + 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(19\)
| 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(103\)
| 103.8.7.1 | $x^{8} + 103$ | $8$ | $1$ | $7$ | $D_{8}$ | $[\ ]_{8}^{2}$ |
103.8.0.1 | $x^{8} + x^{4} + 70 x^{3} + 71 x^{2} + 49 x + 5$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |