Normalized defining polynomial
\( x^{34} + 2 \)
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)
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Discriminant: | \(-100988759218878889187610619236040764506432004552857893115789312\) \(\medspace = -\,2^{67}\cdot 17^{34}\) | 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: | $2^{67/34}17^{287/272}\approx 77.89520567226786$ | ||
Ramified primes: | \(2\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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: | $a^{2}+1$, $a^{18}-a^{2}+1$, $a^{20}-a^{18}-a^{4}+a^{2}-1$, $a^{30}+a^{26}+a^{22}+a^{18}+a^{14}+a^{10}+a^{6}+a^{2}-1$, $a^{33}-a^{32}+a^{30}-a^{29}+a^{27}-a^{26}+a^{24}-a^{23}+a^{21}-a^{20}+a^{18}-a^{17}+a^{15}-a^{14}+a^{12}-a^{11}+a^{9}-a^{8}+a^{6}-a^{5}+a^{3}-a^{2}+1$, $a^{32}+a^{24}-a^{20}-a^{18}+a^{14}+a^{6}+a^{4}-1$, $a^{32}-a^{28}-a^{26}-a^{24}+a^{22}+a^{20}+a^{18}-a^{14}-2a^{12}+a^{8}+2a^{6}+a^{4}-3$, $a^{26}+a^{20}+a^{18}+a^{14}+a^{12}+a^{10}+a^{6}+a^{2}-1$, $a^{32}-a^{30}+a^{28}+a^{24}+a^{22}-a^{20}+a^{18}-2a^{16}+a^{14}-2a^{12}+a^{10}+a^{4}-a^{2}+3$, $a^{33}+a^{30}-2a^{28}-a^{27}-3a^{26}-a^{25}-a^{24}-a^{23}-a^{22}-a^{21}-2a^{20}+a^{19}+a^{18}+a^{17}+3a^{16}+2a^{14}+2a^{13}+2a^{12}+3a^{11}+a^{10}-a^{9}+a^{8}-2a^{7}-a^{5}-2a^{4}-4a^{3}-2a^{2}-5a+1$, $a^{33}-4a^{31}+3a^{29}-2a^{28}+4a^{26}-3a^{25}-4a^{24}+3a^{23}-a^{22}-3a^{21}+5a^{20}+2a^{19}-5a^{18}+2a^{17}+a^{16}-6a^{15}+2a^{14}+5a^{13}-3a^{12}+4a^{10}-4a^{9}-4a^{8}+4a^{7}+a^{6}-2a^{5}+4a^{4}+2a^{3}-5a^{2}-a+3$, $2a^{32}+2a^{31}+2a^{30}-a^{27}-2a^{25}-a^{24}-2a^{23}-a^{21}+2a^{18}+2a^{17}+3a^{16}+a^{15}+a^{14}-2a^{13}-2a^{12}-3a^{11}-a^{10}-a^{9}-a^{7}+a^{6}+a^{5}+3a^{4}+2a^{3}+2a^{2}+2a+1$, $3a^{33}-3a^{32}+2a^{31}-a^{30}-2a^{29}-a^{28}+3a^{27}-5a^{26}+6a^{25}-a^{24}+a^{23}-2a^{22}+4a^{21}-9a^{20}+6a^{19}-6a^{18}+2a^{17}-a^{16}+7a^{15}-7a^{14}+9a^{13}-5a^{12}+a^{11}-2a^{10}+2a^{9}-8a^{8}+9a^{7}-3a^{6}+2a^{5}+3a^{4}+a^{3}-5a^{2}+4a-5$, $2a^{30}-2a^{29}+a^{28}+a^{27}-2a^{26}+4a^{25}-2a^{24}+4a^{22}-5a^{21}+3a^{20}+a^{19}-4a^{18}+5a^{17}-4a^{16}-a^{15}+4a^{14}-4a^{13}+a^{12}-a^{11}-2a^{10}+3a^{9}-a^{7}-a^{6}+a^{5}+a^{4}+a^{3}-2a+3$, $19a^{33}+a^{32}-18a^{31}+9a^{30}+17a^{29}-16a^{28}-11a^{27}+19a^{26}+2a^{25}-22a^{24}+4a^{23}+20a^{22}-11a^{21}-15a^{20}+19a^{19}+13a^{18}-21a^{17}-4a^{16}+26a^{15}-7a^{14}-29a^{13}+15a^{12}+17a^{11}-28a^{10}-10a^{9}+30a^{8}-27a^{6}+13a^{5}+27a^{4}-21a^{3}-21a^{2}+26a+9$, $4a^{33}+12a^{32}+22a^{31}+11a^{30}-2a^{29}-19a^{28}-24a^{27}-7a^{26}+7a^{25}+23a^{24}+19a^{23}+13a^{22}-19a^{21}-24a^{20}-21a^{19}-a^{18}+17a^{17}+28a^{16}+22a^{15}-6a^{14}-21a^{13}-34a^{12}-10a^{11}+3a^{10}+31a^{9}+31a^{8}+15a^{7}-20a^{6}-33a^{5}-26a^{4}-13a^{3}+27a^{2}+36a+33$ (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: | \( 585226461963806300 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{17}\cdot 585226461963806300 \cdot 1}{2\cdot\sqrt{100988759218878889187610619236040764506432004552857893115789312}}\cr\approx \mathstrut & 1.07948149905439 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times F_{17}$ (as 34T9):
A solvable group of order 544 |
The 34 conjugacy class representatives for $C_2\times F_{17}$ |
Character table for $C_2\times F_{17}$ is not computed |
Intermediate fields
\(\Q(\sqrt{-2}) \), 17.1.54214017802982966177103872.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 34 sibling: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | $16^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | $16^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $16^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | $16^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | R | ${\href{/padicField/19.8.0.1}{8} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $16^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $16^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | $16^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | $16^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $16^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{8}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.8.0.1}{8} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.8.0.1}{8} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $34$ | $34$ | $1$ | $67$ | |||
\(17\) | 17.17.17.1 | $x^{17} + 17 x + 17$ | $17$ | $1$ | $17$ | $F_{17}$ | $[17/16]_{16}$ |
17.17.17.1 | $x^{17} + 17 x + 17$ | $17$ | $1$ | $17$ | $F_{17}$ | $[17/16]_{16}$ |