Normalized defining polynomial
\( x^{34} - x + 2 \)
Invariants
Degree: | $34$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 17]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-100988759218749779147522858208201148476497340017318555932408799\) \(\medspace = -\,13\cdot 1061\cdot 192313073533859878065118129\cdot 38\!\cdots\!67\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(66.63\) | 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: | $13^{1/2}1061^{1/2}192313073533859878065118129^{1/2}38071982428796861661094741190767^{1/2}\approx 1.0049316355790068e+31$ | ||
Ramified primes: | \(13\), \(1061\), \(192313073533859878065118129\), \(38071\!\cdots\!90767\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-10098\!\cdots\!08799}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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
Trivial group, which has order $1$ (assuming GRH)
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: | $a^{4}-a^{3}+a^{2}-a+1$, $a^{6}-a^{5}+a^{4}-a^{3}+a^{2}-a+1$, $a^{23}-a^{12}+1$, $a^{32}-a^{30}+a^{28}-a^{26}+a^{24}-a^{22}+a^{20}-a^{18}+a^{16}-a^{14}+a^{12}-a^{10}+a^{8}-a^{6}+a^{4}-a^{2}+1$, $a^{31}+a^{30}+a^{27}+a^{26}+a^{23}+a^{22}+a^{19}+a^{18}+a^{15}+a^{14}+a^{11}+a^{10}+a^{7}+a^{6}-a^{4}+a^{2}-1$, $a^{33}-a^{30}+a^{27}-a^{24}+a^{21}-a^{18}+a^{15}-a^{12}+a^{9}-a^{6}+a^{3}+a^{2}-1$, $a^{29}+a^{28}+a^{27}+a^{26}+a^{25}+a^{24}-a^{21}-a^{19}-a^{17}-a^{15}-a^{13}-a^{12}-a^{11}+a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+1$, $a^{33}-a^{32}+a^{31}+a^{30}+2a^{28}-a^{27}+a^{26}+a^{25}-a^{24}+2a^{23}-2a^{22}+a^{20}-2a^{19}+2a^{18}-2a^{17}-a^{16}+a^{15}-3a^{14}+2a^{13}-a^{12}-a^{11}+2a^{10}-2a^{9}+3a^{8}-a^{6}+2a^{5}-2a^{4}+3a^{3}-a+1$, $a^{33}+2a^{32}+a^{31}+a^{30}+2a^{29}+a^{28}+a^{27}+a^{26}-a^{25}-2a^{24}-2a^{23}-3a^{22}-2a^{21}-a^{20}-2a^{19}-a^{18}-a^{16}+a^{12}+2a^{11}+2a^{10}+2a^{9}+3a^{8}+a^{7}+2a^{6}+2a^{5}-a^{3}-a^{2}-2a-3$, $a^{33}+a^{31}-a^{18}+a^{15}+a^{14}+2a^{12}-a^{11}+a^{10}-2a^{9}-a^{6}+a^{3}+2a^{2}-a+1$, $a^{31}+a^{30}+a^{29}+a^{25}-a^{22}+a^{19}-a^{16}-2a^{15}-a^{14}+2a^{12}-2a^{9}+2a^{8}+a^{7}+2a^{6}-2a^{5}-a^{3}+2a^{2}+1$, $3a^{33}+3a^{32}-a^{31}-4a^{30}-2a^{29}+3a^{27}-4a^{25}-5a^{24}+a^{23}+3a^{22}+2a^{21}-2a^{20}-4a^{19}+5a^{17}+4a^{16}-3a^{14}-2a^{13}+4a^{12}+6a^{11}-4a^{9}-4a^{8}+a^{7}+4a^{6}+3a^{5}-6a^{4}-6a^{3}-a^{2}+4a+1$, $3a^{32}-2a^{31}-2a^{30}+3a^{29}+2a^{28}-4a^{27}+4a^{25}-2a^{24}-3a^{23}+2a^{22}+4a^{21}-4a^{20}-a^{19}+4a^{18}-a^{17}-6a^{16}+a^{15}+6a^{14}-2a^{13}-3a^{12}+4a^{11}+2a^{10}-8a^{9}+a^{8}+6a^{7}-5a^{5}+3a^{4}+4a^{3}-8a^{2}-2a+7$, $6a^{33}+a^{32}-7a^{31}-8a^{30}+7a^{28}+9a^{27}+2a^{26}-8a^{25}-8a^{24}-3a^{23}+4a^{22}+10a^{21}+4a^{20}-5a^{19}-10a^{18}-8a^{17}+5a^{16}+12a^{15}+6a^{14}-2a^{13}-10a^{12}-9a^{11}+2a^{10}+9a^{9}+11a^{8}+3a^{7}-12a^{6}-11a^{5}-a^{4}+9a^{3}+14a^{2}+2a-15$, $3a^{33}+2a^{32}-a^{31}-5a^{30}+2a^{29}+a^{27}-2a^{26}+2a^{25}+2a^{24}+3a^{23}-5a^{22}-4a^{21}+2a^{20}+a^{19}+5a^{18}-4a^{17}+a^{16}+2a^{14}-5a^{13}-a^{12}-a^{11}+6a^{10}+5a^{9}-7a^{8}-4a^{6}+6a^{5}-3a^{4}-a^{3}-2a^{2}+10a-5$, $a^{33}-3a^{32}+3a^{31}-2a^{30}-2a^{28}+2a^{27}-4a^{26}+2a^{25}-4a^{24}+a^{23}-a^{22}+3a^{21}-5a^{20}+4a^{19}-3a^{18}+a^{17}-2a^{16}+a^{15}-4a^{14}+4a^{13}-a^{12}-a^{11}+a^{10}+a^{9}-a^{8}+2a^{7}-2a^{6}-3a^{5}+5a^{4}-2a^{3}+a^{2}+3$ (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: | \( 606779702973969700 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{17}\cdot 606779702973969700 \cdot 1}{2\cdot\sqrt{100988759218749779147522858208201148476497340017318555932408799}}\cr\approx \mathstrut & 1.11923760447292 \end{aligned}\] (assuming GRH)
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 | ${\href{/padicField/2.10.0.1}{10} }^{3}{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $30{,}\,{\href{/padicField/5.4.0.1}{4} }$ | $16{,}\,{\href{/padicField/7.9.0.1}{9} }^{2}$ | ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | $16{,}\,{\href{/padicField/17.13.0.1}{13} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | $19{,}\,15$ | $16{,}\,{\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.8.0.1}{8} }$ | $18{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $31{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $32{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $28{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.9.0.1}{9} }^{2}{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/59.13.0.1}{13} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.11.0.1 | $x^{11} + 3 x + 11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
13.17.0.1 | $x^{17} + 10 x^{2} + 6 x + 11$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
\(1061\) | $\Q_{1061}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $27$ | $1$ | $27$ | $0$ | $C_{27}$ | $[\ ]^{27}$ | ||
\(192\!\cdots\!129\) | $\Q_{19\!\cdots\!29}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $28$ | $1$ | $28$ | $0$ | $C_{28}$ | $[\ ]^{28}$ | ||
\(380\!\cdots\!767\) | $\Q_{38\!\cdots\!67}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |