Normalized defining polynomial
\( x^{16} + 194 x^{14} + 3183 x^{12} - 109236 x^{10} - 2269931 x^{8} - 11265729 x^{6} + 426352493 x^{4} + \cdots + 41151773881 \)
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: | \(22245513193027862267173879060120840909081\) \(\medspace = 61^{14}\cdot 83^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(332.43\) | 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: | $61^{7/8}83^{1/2}\approx 332.4318694032887$ | ||
Ramified primes: | \(61\), \(83\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10}a^{8}+\frac{3}{10}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{3}{10}a^{2}-\frac{1}{2}a-\frac{1}{10}$, $\frac{1}{10}a^{9}+\frac{3}{10}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{3}{10}a^{3}-\frac{1}{2}a^{2}-\frac{1}{10}a$, $\frac{1}{10}a^{10}-\frac{2}{5}a^{6}-\frac{1}{2}a^{5}-\frac{1}{5}a^{4}-\frac{1}{2}a+\frac{3}{10}$, $\frac{1}{10}a^{11}-\frac{2}{5}a^{7}-\frac{1}{2}a^{6}-\frac{1}{5}a^{5}-\frac{1}{2}a^{2}+\frac{3}{10}a$, $\frac{1}{650}a^{12}-\frac{16}{325}a^{10}+\frac{1}{25}a^{8}-\frac{1}{2}a^{7}-\frac{142}{325}a^{6}-\frac{103}{325}a^{4}-\frac{1}{2}a^{3}-\frac{177}{650}a^{2}+\frac{22}{325}$, $\frac{1}{650}a^{13}-\frac{16}{325}a^{11}+\frac{1}{25}a^{9}-\frac{142}{325}a^{7}-\frac{1}{2}a^{6}-\frac{103}{325}a^{5}+\frac{74}{325}a^{3}-\frac{1}{2}a^{2}-\frac{281}{650}a-\frac{1}{2}$, $\frac{1}{72\!\cdots\!50}a^{14}+\frac{34\!\cdots\!41}{72\!\cdots\!50}a^{12}+\frac{42\!\cdots\!67}{14\!\cdots\!50}a^{10}-\frac{40\!\cdots\!08}{36\!\cdots\!75}a^{8}+\frac{33\!\cdots\!96}{36\!\cdots\!75}a^{6}-\frac{1}{2}a^{5}+\frac{18\!\cdots\!27}{72\!\cdots\!75}a^{4}-\frac{15\!\cdots\!21}{36\!\cdots\!75}a^{2}-\frac{20\!\cdots\!23}{72\!\cdots\!50}$, $\frac{1}{14\!\cdots\!50}a^{15}-\frac{95\!\cdots\!17}{73\!\cdots\!25}a^{13}+\frac{69\!\cdots\!77}{29\!\cdots\!50}a^{11}-\frac{36\!\cdots\!08}{73\!\cdots\!25}a^{9}+\frac{23\!\cdots\!21}{73\!\cdots\!25}a^{7}+\frac{35\!\cdots\!67}{14\!\cdots\!25}a^{5}+\frac{10\!\cdots\!29}{73\!\cdots\!25}a^{3}-\frac{1}{2}a^{2}+\frac{23\!\cdots\!26}{73\!\cdots\!25}a-\frac{1}{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}\times C_{18}$, which has order $54$ (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: | $\frac{2590050292832}{13\!\cdots\!25}a^{14}+\frac{475696298397262}{13\!\cdots\!25}a^{12}+\frac{677752084436214}{27\!\cdots\!25}a^{10}-\frac{30\!\cdots\!37}{13\!\cdots\!25}a^{8}-\frac{25\!\cdots\!81}{13\!\cdots\!25}a^{6}-\frac{21\!\cdots\!12}{27\!\cdots\!25}a^{4}+\frac{10\!\cdots\!81}{13\!\cdots\!25}a^{2}+\frac{14\!\cdots\!14}{13\!\cdots\!25}$, $\frac{64\!\cdots\!78}{55\!\cdots\!75}a^{15}+\frac{12\!\cdots\!98}{55\!\cdots\!75}a^{13}+\frac{27\!\cdots\!61}{11\!\cdots\!75}a^{11}-\frac{97\!\cdots\!23}{55\!\cdots\!75}a^{9}-\frac{11\!\cdots\!49}{55\!\cdots\!75}a^{7}+\frac{32\!\cdots\!67}{11\!\cdots\!75}a^{5}+\frac{26\!\cdots\!49}{55\!\cdots\!75}a^{3}+\frac{91\!\cdots\!81}{55\!\cdots\!75}a$, $\frac{24\!\cdots\!83}{55\!\cdots\!75}a^{15}+\frac{51\!\cdots\!28}{55\!\cdots\!75}a^{13}+\frac{37\!\cdots\!36}{11\!\cdots\!75}a^{11}+\frac{13\!\cdots\!47}{55\!\cdots\!75}a^{9}-\frac{28\!\cdots\!14}{55\!\cdots\!75}a^{7}-\frac{19\!\cdots\!43}{11\!\cdots\!75}a^{5}-\frac{11\!\cdots\!61}{55\!\cdots\!75}a^{3}-\frac{44\!\cdots\!59}{55\!\cdots\!75}a$, $\frac{47\!\cdots\!39}{11\!\cdots\!50}a^{15}-\frac{34\!\cdots\!88}{36\!\cdots\!75}a^{14}+\frac{57\!\cdots\!31}{73\!\cdots\!25}a^{13}-\frac{55\!\cdots\!83}{36\!\cdots\!75}a^{12}+\frac{27\!\cdots\!09}{29\!\cdots\!50}a^{11}+\frac{11\!\cdots\!04}{72\!\cdots\!75}a^{10}-\frac{57\!\cdots\!49}{11\!\cdots\!50}a^{9}+\frac{81\!\cdots\!58}{36\!\cdots\!75}a^{8}-\frac{10\!\cdots\!31}{14\!\cdots\!50}a^{7}-\frac{61\!\cdots\!71}{36\!\cdots\!75}a^{6}-\frac{11\!\cdots\!27}{29\!\cdots\!50}a^{5}+\frac{95\!\cdots\!17}{11\!\cdots\!50}a^{4}+\frac{31\!\cdots\!31}{14\!\cdots\!50}a^{3}-\frac{78\!\cdots\!29}{36\!\cdots\!75}a^{2}+\frac{33\!\cdots\!39}{14\!\cdots\!50}a-\frac{11\!\cdots\!27}{72\!\cdots\!50}$, $\frac{17\!\cdots\!23}{73\!\cdots\!25}a^{15}-\frac{12\!\cdots\!77}{72\!\cdots\!50}a^{14}+\frac{25\!\cdots\!43}{73\!\cdots\!25}a^{13}-\frac{13\!\cdots\!41}{36\!\cdots\!75}a^{12}-\frac{19\!\cdots\!94}{14\!\cdots\!25}a^{11}-\frac{80\!\cdots\!72}{72\!\cdots\!75}a^{10}-\frac{77\!\cdots\!11}{14\!\cdots\!50}a^{9}+\frac{17\!\cdots\!07}{72\!\cdots\!50}a^{8}+\frac{93\!\cdots\!91}{73\!\cdots\!25}a^{7}+\frac{67\!\cdots\!41}{72\!\cdots\!50}a^{6}+\frac{51\!\cdots\!99}{29\!\cdots\!50}a^{5}-\frac{15\!\cdots\!69}{72\!\cdots\!75}a^{4}-\frac{36\!\cdots\!57}{14\!\cdots\!50}a^{3}-\frac{13\!\cdots\!41}{72\!\cdots\!50}a^{2}-\frac{33\!\cdots\!83}{14\!\cdots\!50}a-\frac{74\!\cdots\!29}{72\!\cdots\!50}$, $\frac{16\!\cdots\!11}{55\!\cdots\!75}a^{15}-\frac{48\!\cdots\!77}{36\!\cdots\!75}a^{14}+\frac{69\!\cdots\!77}{11\!\cdots\!50}a^{13}-\frac{10\!\cdots\!57}{36\!\cdots\!75}a^{12}+\frac{50\!\cdots\!59}{22\!\cdots\!50}a^{11}-\frac{74\!\cdots\!64}{72\!\cdots\!75}a^{10}+\frac{89\!\cdots\!24}{55\!\cdots\!75}a^{9}-\frac{25\!\cdots\!68}{36\!\cdots\!75}a^{8}-\frac{19\!\cdots\!13}{55\!\cdots\!75}a^{7}+\frac{60\!\cdots\!16}{36\!\cdots\!75}a^{6}-\frac{12\!\cdots\!91}{11\!\cdots\!75}a^{5}+\frac{39\!\cdots\!77}{72\!\cdots\!75}a^{4}-\frac{75\!\cdots\!37}{55\!\cdots\!75}a^{3}+\frac{23\!\cdots\!34}{36\!\cdots\!75}a^{2}-\frac{29\!\cdots\!28}{55\!\cdots\!75}a+\frac{18\!\cdots\!17}{72\!\cdots\!50}$, $\frac{61\!\cdots\!73}{73\!\cdots\!25}a^{15}+\frac{11\!\cdots\!43}{73\!\cdots\!25}a^{13}+\frac{13\!\cdots\!46}{14\!\cdots\!25}a^{11}-\frac{74\!\cdots\!68}{73\!\cdots\!25}a^{9}-\frac{53\!\cdots\!59}{73\!\cdots\!25}a^{7}-\frac{14\!\cdots\!68}{14\!\cdots\!25}a^{5}+\frac{26\!\cdots\!59}{73\!\cdots\!25}a^{3}+\frac{22\!\cdots\!96}{73\!\cdots\!25}a$ (assuming GRH) | 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: | \( 432906818173 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 432906818173 \cdot 54}{2\cdot\sqrt{22245513193027862267173879060120840909081}}\cr\approx \mathstrut & 0.190360099153685 \end{aligned}\] (assuming GRH)
Galois group
$\OD_{16}:C_2$ (as 16T36):
A solvable group of order 32 |
The 11 conjugacy class representatives for $\OD_{16}:C_2$ |
Character table for $\OD_{16}:C_2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.2.0.1}{2} }^{8}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(61\) | 61.8.7.2 | $x^{8} + 61$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ |
61.8.7.2 | $x^{8} + 61$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $[\ ]_{8}^{2}$ | |
\(83\) | 83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.4.2.1 | $x^{4} + 164 x^{3} + 6894 x^{2} + 13940 x + 564653$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
83.4.2.1 | $x^{4} + 164 x^{3} + 6894 x^{2} + 13940 x + 564653$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |