Normalized defining polynomial
\( x^{34} - 2 \)
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: | \(100988759218878889187610619236040764506432004552857893115789312\) \(\medspace = 2^{67}\cdot 17^{34}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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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: | $17$ | 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: | $a^{17}-1$, $a+1$, $a-1$, $a^{18}-a^{2}-1$, $a^{20}+a^{18}-a^{4}-a^{2}-1$, $a^{30}+a^{26}+a^{22}+a^{18}+a^{14}+a^{10}+a^{6}+a^{2}+1$, $a^{32}+a^{24}-a^{20}+a^{18}-a^{14}-a^{6}+a^{4}-1$, $a^{32}+a^{30}+a^{28}-a^{22}-a^{18}-a^{14}-a^{10}-a^{6}+1$, $a^{26}-a^{20}+a^{18}+a^{14}-a^{12}+a^{10}+a^{6}+a^{2}+1$, $a^{32}+a^{30}-a^{22}+a^{18}-a^{14}+a^{12}+a^{10}-2a^{2}-1$, $a^{33}+a^{32}+a^{30}-a^{27}-a^{26}-a^{25}-a^{24}-a^{23}-a^{22}+a^{18}+a^{17}+2a^{16}+a^{15}+a^{14}+a^{13}-a^{12}+a^{11}-2a^{10}-a^{9}-a^{8}-2a^{7}-a^{6}-2a^{5}+a^{3}+2a+1$, $a^{31}-a^{30}-a^{29}-a^{28}+2a^{26}-a^{25}+a^{22}+2a^{21}-a^{20}-2a^{19}-a^{18}+a^{17}+a^{15}-3a^{14}+a^{13}+2a^{12}+a^{11}+a^{10}-3a^{9}+a^{6}-2a^{5}-a^{4}+2a^{2}+2a-1$, $2a^{33}-a^{31}+a^{30}+a^{29}-2a^{27}+a^{26}+a^{25}-2a^{24}-a^{23}+a^{21}-2a^{20}-a^{19}+2a^{18}-a^{16}+3a^{14}-2a^{12}+2a^{11}+a^{10}-a^{9}-2a^{8}+2a^{6}-4a^{5}-a^{4}+a^{3}+a^{2}-2a-1$, $a^{33}+a^{32}-a^{30}-a^{29}-a^{28}-a^{24}-a^{23}+a^{21}+a^{20}-a^{19}-2a^{18}-2a^{17}+2a^{15}+3a^{14}+2a^{13}-a^{12}-2a^{11}-2a^{10}+a^{8}+3a^{7}+3a^{6}+a^{5}-a^{4}-2a^{3}-a^{2}-a+1$, $10a^{33}+16a^{32}-13a^{31}-13a^{30}+16a^{29}+11a^{28}-17a^{27}-9a^{26}+21a^{25}+6a^{24}-23a^{23}-a^{22}+25a^{21}-4a^{20}-24a^{19}+10a^{18}+22a^{17}-11a^{16}-22a^{15}+17a^{14}+21a^{13}-22a^{12}-17a^{11}+28a^{10}+12a^{9}-30a^{8}-5a^{7}+31a^{6}+2a^{5}-32a^{4}+4a^{3}+35a^{2}-10a-35$, $9a^{33}+6a^{32}-a^{31}-6a^{30}-11a^{29}-9a^{28}-9a^{27}-2a^{26}+a^{25}+9a^{24}+10a^{23}+12a^{22}+5a^{21}+a^{20}-5a^{19}-8a^{18}-10a^{17}-10a^{16}-6a^{15}-a^{14}+9a^{13}+10a^{12}+14a^{11}+8a^{10}+7a^{9}-4a^{8}-7a^{7}-18a^{6}-15a^{5}-13a^{4}-a^{3}+6a^{2}+16a+19$, $6a^{33}-6a^{32}-5a^{31}-5a^{30}-4a^{29}+10a^{28}+10a^{27}-a^{26}-5a^{25}-6a^{24}-8a^{23}+4a^{22}+11a^{21}+4a^{20}-a^{19}-14a^{17}-5a^{16}+7a^{15}+8a^{14}+8a^{13}+4a^{12}-12a^{11}-16a^{10}+3a^{9}+7a^{8}+9a^{7}+12a^{6}-5a^{5}-17a^{4}-3a^{3}-3a^{2}+5a+15$ (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: | \( 504347647410496830 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{16}\cdot 504347647410496830 \cdot 1}{2\cdot\sqrt{100988759218878889187610619236040764506432004552857893115789312}}\cr\approx \mathstrut & 0.592244955912632 \end{aligned}\] (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 $C_2\times F_{17}$ |
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 | $16^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $16^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $16^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $16^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }^{8}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | R | ${\href{/padicField/19.8.0.1}{8} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | $16^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $16^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | $16^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $16^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $16^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.8.0.1}{8} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 | $x^{17} + 17 x + 17$ | $17$ | $1$ | $17$ | $F_{17}$ | $[17/16]_{16}$ |