Normalized defining polynomial
\( x^{34} - x + 3 \)
Invariants
Degree: | $34$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 17]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-65355867734067162148565103624914668628139062779670233016609989966815\) \(\medspace = -\,3^{33}\cdot 5\cdot 7\cdot 89\cdot 97\cdot 397\cdot 98\!\cdots\!83\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(98.76\) | 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: | \(3\), \(5\), \(7\), \(89\), \(97\), \(397\), \(98008\!\cdots\!18683\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-35269\!\cdots\!81215}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $16$ | 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: | not computed | 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: | not computed | 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 $
Galois group
A non-solvable group of order 295232799039604140847618609643520000000 |
The 12310 conjugacy class representatives for $S_{34}$ are not computed |
Character table for $S_{34}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $30{,}\,{\href{/padicField/2.4.0.1}{4} }$ | R | R | R | $34$ | $29{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | $19{,}\,15$ | $16{,}\,{\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.8.0.1}{8} }$ | $34$ | $32{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ | $19{,}\,{\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.11.0.1}{11} }^{2}{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | $24{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.3.3.1 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ | |
3.15.15.34 | $x^{15} - 18 x^{14} + 462 x^{13} + 3948 x^{12} + 46782 x^{11} + 173079 x^{10} + 158742 x^{9} + 124092 x^{8} + 152685 x^{7} + 212733 x^{6} + 149931 x^{5} + 90234 x^{4} + 37827 x^{3} + 12150 x^{2} + 2430 x + 243$ | $3$ | $5$ | $15$ | 15T44 | $[3/2, 3/2, 3/2, 3/2, 3/2]_{2}^{5}$ | |
3.15.15.44 | $x^{15} + 6 x^{14} + 27 x^{13} + 294 x^{12} + 819 x^{11} + 1080 x^{10} + 144 x^{9} + 1863 x^{8} + 1296 x^{7} + 2781 x^{6} + 648 x^{5} + 3402 x^{4} - 1053 x^{3} + 972 x^{2} - 729 x + 243$ | $3$ | $5$ | $15$ | 15T44 | $[3/2, 3/2, 3/2, 3/2, 3/2]_{2}^{5}$ | |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.13.0.1 | $x^{13} + 4 x^{2} + 3 x + 3$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
5.18.0.1 | $x^{18} + x^{12} + x^{11} + x^{10} + x^{9} + 2 x^{8} + 2 x^{6} + x^{5} + 2 x^{3} + 2 x^{2} + 2$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.15.0.1 | $x^{15} + 5 x^{6} + 6 x^{5} + 6 x^{4} + 4 x^{3} + x^{2} + 2 x + 4$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
7.15.0.1 | $x^{15} + 5 x^{6} + 6 x^{5} + 6 x^{4} + 4 x^{3} + x^{2} + 2 x + 4$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
\(89\) | $\Q_{89}$ | $x + 86$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{89}$ | $x + 86$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
89.2.1.1 | $x^{2} + 89$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
89.6.0.1 | $x^{6} + x^{4} + 82 x^{3} + 80 x^{2} + 15 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ | ||
\(97\) | $\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
97.2.1.2 | $x^{2} + 485$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $30$ | $1$ | $30$ | $0$ | $C_{30}$ | $[\ ]^{30}$ | ||
\(397\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $28$ | $1$ | $28$ | $0$ | $C_{28}$ | $[\ ]^{28}$ | ||
\(980\!\cdots\!683\) | $\Q_{98\!\cdots\!83}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $29$ | $1$ | $29$ | $0$ | $C_{29}$ | $[\ ]^{29}$ |