Normalized defining polynomial
\( x^{34} - 2x - 1 \)
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)
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Discriminant: | \(2218093610805369243106601538481015288800287251162174144380928\) \(\medspace = 2^{35}\cdot 307\cdot 135071166289\cdot 15\!\cdots\!27\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(59.55\) | 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: | \(2\), \(307\), \(135071166289\), \(15567\!\cdots\!82827\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{12911\!\cdots\!67842}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $17$ | 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: | $a$, $a^{33}-a^{16}-1$, $a^{33}+a-1$, $a^{22}+a^{11}+1$, $a^{33}-a^{32}+a^{31}-a^{30}+a^{29}-a^{28}+a^{27}-a^{26}+a^{22}-a^{21}+a^{20}-a^{19}+a^{18}-a^{17}+a^{16}-a^{15}+a^{11}-a^{10}+a^{9}-a^{8}+a^{7}-a^{6}+a^{5}-a^{4}-1$, $a^{31}-a^{29}+a^{27}-a^{25}+a^{23}-a^{21}+a^{19}-a^{17}+a^{15}-a^{13}+a^{11}+a^{10}-a^{9}-2a^{8}+a^{7}+2a^{6}-a^{5}-2a^{4}+a^{3}+2a^{2}-a-2$, $a^{30}+a^{26}-a^{25}+a^{24}+a^{22}-a^{21}+2a^{18}-a^{17}-a^{15}+a^{14}-a^{11}+a^{10}-a^{9}+a^{6}-a^{4}-a^{3}+2a^{2}+a-1$, $2a^{33}+2a^{31}+a^{28}-a^{27}-a^{24}-2a^{21}-2a^{19}-2a^{18}-a^{17}-3a^{16}-2a^{15}-a^{14}-2a^{13}+a^{11}-a^{10}+2a^{9}+2a^{6}+a^{4}+3a^{3}+2a^{2}+4a+1$, $a^{33}+2a^{32}+a^{31}-a^{29}-a^{28}-a^{27}+a^{25}+2a^{24}+2a^{23}+a^{22}-a^{21}-2a^{20}-2a^{19}-a^{18}+a^{17}+2a^{16}+2a^{15}+a^{14}-a^{13}-3a^{12}-3a^{11}-2a^{10}+2a^{8}+2a^{7}+2a^{6}-2a^{4}-4a^{3}-3a^{2}-a+1$, $a^{31}-a^{30}+a^{28}-2a^{27}+a^{26}+a^{24}-2a^{23}+2a^{22}-a^{21}-a^{20}+a^{19}-a^{16}+2a^{15}-3a^{14}+a^{13}+a^{11}-a^{10}+a^{9}-2a^{7}+a^{6}+a^{4}-2a^{3}+2a^{2}-2a-1$, $a^{30}-a^{29}-a^{27}+a^{26}+a^{24}-a^{23}-a^{21}+a^{20}+a^{18}-a^{17}+a^{14}-a^{11}+a^{9}+a^{8}-a^{6}-a^{5}-a^{4}+a^{3}-1$, $a^{31}-a^{30}+a^{29}+a^{28}+a^{26}+a^{24}-a^{23}+a^{22}+a^{21}-a^{20}+2a^{19}+a^{17}+a^{15}+a^{14}-a^{13}+2a^{12}+a^{9}+a^{8}+a^{7}+2a^{5}+2a^{2}+a+1$, $a^{33}-a^{32}+a^{31}-a^{30}-a^{28}-a^{25}-a^{21}+a^{20}-a^{19}+a^{18}-a^{17}-a^{12}+2a^{11}-a^{10}+a^{9}+2a^{7}+a^{6}+a^{5}+a^{3}+3a^{2}-a-2$, $3a^{33}-3a^{32}+2a^{31}-a^{29}+3a^{28}-4a^{27}+2a^{26}-2a^{25}-3a^{22}+4a^{21}-3a^{20}+3a^{19}-a^{18}-a^{17}+2a^{16}-4a^{15}+3a^{14}-5a^{13}+2a^{12}-a^{11}-a^{10}+3a^{9}-3a^{8}+3a^{7}-2a^{6}+a^{5}-5a^{3}+3a^{2}-5a-3$, $a^{33}+a^{32}-2a^{31}-a^{30}+3a^{29}-3a^{28}-3a^{27}+3a^{26}+a^{25}-4a^{24}+3a^{22}-3a^{21}+3a^{19}-2a^{18}-3a^{17}+5a^{16}+a^{15}-5a^{14}+2a^{13}+4a^{12}-2a^{11}-a^{10}+5a^{9}-2a^{8}-a^{7}+6a^{6}+a^{5}-6a^{4}+3a^{3}+6a^{2}-5a-4$, $a^{33}-a^{32}+2a^{31}-3a^{30}+a^{29}-2a^{28}+2a^{27}+a^{26}+a^{25}-2a^{24}-2a^{22}+a^{21}+a^{19}+a^{18}-a^{17}-2a^{16}-a^{15}+2a^{14}+a^{13}+2a^{12}-2a^{11}-a^{10}-2a^{9}-a^{7}+5a^{6}-a^{5}+a^{4}-4a^{3}-a^{2}+1$, $2a^{33}+3a^{32}-10a^{31}+3a^{30}+6a^{29}+a^{28}-7a^{27}-3a^{26}+10a^{25}-a^{24}-4a^{23}-7a^{22}+11a^{21}+2a^{20}-5a^{19}-7a^{18}+7a^{17}+8a^{16}-9a^{15}-6a^{14}+2a^{13}+13a^{12}-9a^{11}-7a^{10}+2a^{9}+14a^{8}-3a^{7}-12a^{6}+3a^{5}+10a^{4}+3a^{3}-18a^{2}+a+6$ (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: | \( 65073117810876920 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{16}\cdot 65073117810876920 \cdot 1}{2\cdot\sqrt{2218093610805369243106601538481015288800287251162174144380928}}\cr\approx \mathstrut & 0.515607907637809 \end{aligned}\] (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 | R | $17{,}\,{\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $34$ | $33{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $28{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | $21{,}\,{\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $25{,}\,{\href{/padicField/37.9.0.1}{9} }$ | $34$ | $28{,}\,{\href{/padicField/43.6.0.1}{6} }$ | $34$ | $26{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.1 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
Deg $16$ | $2$ | $8$ | $16$ | ||||
Deg $16$ | $2$ | $8$ | $16$ | ||||
\(307\) | $\Q_{307}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{307}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(135071166289\) | $\Q_{135071166289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
\(155\!\cdots\!827\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |