Normalized defining polynomial
\( x^{34} - 2x - 2 \)
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: | \(103206852817927619525348604763107729293921936249402125350600704\) \(\medspace = 2^{34}\cdot 371237\cdot 768749632105135909\cdot 21050019716166134869652117257\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(66.67\) | 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: | \(2\), \(371237\), \(768749632105135909\), \(21050019716166134869652117257\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{60074\!\cdots\!65281}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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)
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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: | $a+1$, $a^{4}-a^{2}+1$, $a^{23}+a^{12}+a+1$, $a^{17}-a-1$, $a^{15}-a^{14}+a^{12}-a^{11}+a^{9}-a^{8}+a^{6}-a^{5}+a^{3}-a^{2}+1$, $a^{32}-a^{31}+a^{29}-a^{28}+a^{25}-2a^{24}+2a^{23}-a^{20}+2a^{19}-a^{18}-a^{17}+a^{16}-a^{14}+a^{12}-2a^{11}+a^{9}-a^{8}-a^{7}+2a^{6}+a^{5}-2a^{4}+a^{3}+2a^{2}-2a-1$, $2a^{33}-a^{32}-a^{27}+a^{26}-2a^{25}+2a^{24}-2a^{23}-a^{20}+a^{19}-a^{18}-a^{16}+2a^{15}-2a^{14}+2a^{13}-a^{12}+2a^{10}-a^{9}+a^{8}+a^{6}+2a^{4}-2a^{3}+a^{2}+a-5$, $8a^{33}-10a^{32}+10a^{31}-9a^{30}+7a^{29}-3a^{28}-2a^{27}+6a^{26}-8a^{25}+9a^{24}-9a^{23}+7a^{22}-5a^{21}+4a^{20}-3a^{19}+2a^{18}-2a^{16}+4a^{15}-8a^{14}+10a^{13}-9a^{12}+8a^{11}-5a^{10}+a^{9}+3a^{8}-6a^{7}+5a^{6}-5a^{5}+6a^{4}-4a^{3}+3a^{2}-4a-11$, $5a^{33}-3a^{32}+2a^{30}-3a^{29}+3a^{28}-a^{27}+2a^{25}-2a^{24}+2a^{22}-4a^{21}+5a^{20}-3a^{19}+2a^{18}+a^{17}-2a^{16}+a^{15}-3a^{13}+5a^{12}-4a^{11}+5a^{10}-a^{9}-a^{8}+3a^{7}-4a^{6}+a^{5}+a^{4}-2a^{3}+4a^{2}-11$, $a^{33}-a^{32}+a^{31}-2a^{30}+3a^{29}-2a^{28}+2a^{27}-2a^{26}+a^{25}-3a^{24}+3a^{23}+2a^{21}-2a^{20}-a^{19}-2a^{18}+2a^{17}+2a^{16}+a^{15}-a^{14}-3a^{13}+a^{11}+2a^{10}+a^{8}-3a^{7}+a^{4}-a^{3}+3a^{2}-a-3$, $a^{33}-a^{30}-a^{28}-a^{26}+a^{25}+a^{23}+a^{21}-a^{18}-a^{15}+2a^{13}+a^{12}+2a^{10}+2a^{9}-2a^{8}-a^{7}-2a^{5}-3a^{4}+a^{3}+a^{2}-1$, $a^{32}-a^{31}+a^{30}+a^{26}+a^{23}+a^{19}-a^{18}-a^{14}+a^{13}+a^{9}-a^{8}+a^{6}-a^{4}+a^{3}+a^{2}-a+1$, $a^{33}+a^{32}+2a^{31}+2a^{30}+a^{28}-a^{26}-3a^{24}-3a^{23}-4a^{21}+a^{19}-2a^{18}+3a^{17}+2a^{16}+2a^{15}+6a^{14}+2a^{12}+3a^{11}-3a^{10}-2a^{8}-6a^{7}-3a^{6}-5a^{5}-3a^{4}-2a^{2}+a+3$, $a^{33}+4a^{32}-2a^{31}-3a^{30}+4a^{29}+2a^{28}-4a^{27}+4a^{25}-2a^{24}-3a^{23}+2a^{22}+a^{21}-3a^{20}-a^{19}+2a^{18}-a^{16}+a^{14}+2a^{13}-2a^{11}+2a^{10}+3a^{9}-3a^{8}-4a^{7}+5a^{6}-8a^{4}+a^{3}+7a^{2}-6a-9$, $2a^{33}-a^{30}+a^{29}+a^{28}-2a^{26}-2a^{25}+a^{24}+2a^{23}-a^{21}+2a^{19}+a^{18}-a^{17}-3a^{16}-a^{15}+2a^{14}+2a^{13}-2a^{12}-2a^{11}+a^{10}+3a^{9}+a^{8}-2a^{7}-2a^{6}+a^{5}+3a^{4}-5a^{2}-3a-1$, $2a^{33}-2a^{32}+2a^{31}-2a^{30}+3a^{29}-2a^{28}+3a^{27}-3a^{26}+3a^{25}-3a^{24}+2a^{23}-2a^{22}+a^{21}-a^{19}+a^{18}-3a^{17}+3a^{16}-4a^{15}+4a^{14}-3a^{13}+5a^{12}-2a^{11}+3a^{10}-2a^{9}+a^{8}-a^{7}-a^{6}-a^{4}+a^{3}-2a^{2}-5$, $a^{32}-a^{31}-a^{30}-a^{29}+a^{26}+a^{24}-a^{19}+a^{16}+a^{15}-a^{13}-2a^{12}-3a^{11}+2a^{9}+3a^{8}+3a^{7}+a^{6}-2a^{5}-3a^{4}-3a^{3}+3a+3$ (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: | \( 612915422175067900 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{16}\cdot 612915422175067900 \cdot 1}{2\cdot\sqrt{103206852817927619525348604763107729293921936249402125350600704}}\cr\approx \mathstrut & 0.711957666816501 \end{aligned}\] (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 $S_{34}$ 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 | R | $28{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $30{,}\,{\href{/padicField/5.4.0.1}{4} }$ | $34$ | $34$ | $21{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $26{,}\,{\href{/padicField/17.8.0.1}{8} }$ | $15{,}\,{\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | $16{,}\,{\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $34$ | $17{,}\,{\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $34$ | $34$ | $1$ | $34$ | |||
\(371237\) | $\Q_{371237}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
\(768749632105135909\) | $\Q_{768749632105135909}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{768749632105135909}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{768749632105135909}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $25$ | $1$ | $25$ | $0$ | $C_{25}$ | $[\ ]^{25}$ | ||
\(210\!\cdots\!257\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ |