Normalized defining polynomial
\( x^{34} - 5x + 3 \)
Invariants
Degree: | $34$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(8350207664850893587674765141294032045697120106790441256127699394134602033\) \(\medspace = 3^{31}\cdot 2879\cdot 451547\cdot 77705103841531489\cdot 13\!\cdots\!27\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(139.56\) | 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: | $3^{241/162}2879^{1/2}451547^{1/2}77705103841531489^{1/2}133827143657449261027156958311927^{1/2}\approx 5.9601979834645525e+29$ | ||
Ramified primes: | \(3\), \(2879\), \(451547\), \(77705103841531489\), \(13382\!\cdots\!11927\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{40556\!\cdots\!57017}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{3}a^{33}-\frac{1}{3}a^{32}+\frac{1}{3}a^{31}-\frac{1}{3}a^{30}+\frac{1}{3}a^{29}-\frac{1}{3}a^{28}+\frac{1}{3}a^{27}-\frac{1}{3}a^{26}+\frac{1}{3}a^{25}-\frac{1}{3}a^{24}+\frac{1}{3}a^{23}-\frac{1}{3}a^{22}+\frac{1}{3}a^{21}-\frac{1}{3}a^{20}+\frac{1}{3}a^{19}-\frac{1}{3}a^{18}+\frac{1}{3}a^{17}-\frac{1}{3}a^{16}+\frac{1}{3}a^{15}-\frac{1}{3}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $17$ | 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 $S_{34}$ is not computed |
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 | $16^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $34$ | $31{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }$ | $17{,}\,15{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $25{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $33{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/41.13.0.1}{13} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $26{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | $29{,}\,{\href{/padicField/47.1.0.1}{1} }^{5}$ | $16{,}\,{\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.15.15.12 | $x^{15} + 30 x^{14} + 567 x^{13} + 5343 x^{12} + 26946 x^{11} + 81891 x^{10} + 187308 x^{9} + 338094 x^{8} + 444204 x^{7} + 428112 x^{6} + 240165 x^{5} + 86994 x^{4} - 10773 x^{3} + 15309 x^{2} - 1458 x + 243$ | $3$ | $5$ | $15$ | 15T33 | $[3/2, 3/2, 3/2, 3/2]_{2}^{5}$ | |
3.15.15.12 | $x^{15} + 30 x^{14} + 567 x^{13} + 5343 x^{12} + 26946 x^{11} + 81891 x^{10} + 187308 x^{9} + 338094 x^{8} + 444204 x^{7} + 428112 x^{6} + 240165 x^{5} + 86994 x^{4} - 10773 x^{3} + 15309 x^{2} - 1458 x + 243$ | $3$ | $5$ | $15$ | 15T33 | $[3/2, 3/2, 3/2, 3/2]_{2}^{5}$ | |
\(2879\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | ||
\(451547\) | $\Q_{451547}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{451547}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{451547}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | ||
\(77705103841531489\) | $\Q_{77705103841531489}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $29$ | $1$ | $29$ | $0$ | $C_{29}$ | $[\ ]^{29}$ | ||
\(133\!\cdots\!927\) | $\Q_{13\!\cdots\!27}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ |