Normalized defining polynomial
\( x^{34} - x - 1 \)
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)
| |
Discriminant: | \(11885748945456377039253666531245020090157279092950049\) \(\medspace = 7\cdot 191\cdot 607\cdot 14\!\cdots\!11\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(34.01\) | 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: | $7^{1/2}191^{1/2}607^{1/2}14645575916792712592989131451003587034531413111^{1/2}\approx 1.0902178197707272e+26$ | ||
Ramified primes: | \(7\), \(191\), \(607\), \(14645\!\cdots\!13111\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{11885\!\cdots\!50049}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a^{33}-1$, $a^{33}-a^{32}-a^{15}-1$, $a^{14}-a^{3}$, $a^{21}-a^{20}$, $a^{33}-a^{32}+a^{31}-a^{30}+a^{29}-a^{28}+a^{27}-a^{26}+a^{25}-a^{24}-a^{22}-1$, $a^{33}-a^{32}+a^{31}-a^{30}+a^{29}-a^{28}+a^{27}$, $a^{31}-a^{30}+a^{29}-a^{28}+a^{27}$, $a^{33}-a^{32}+a^{31}-a^{30}+a^{27}-a^{26}-1$, $a^{12}-a^{9}+a^{6}$, $a^{33}-a^{32}+a^{31}-a^{30}+a^{29}-a^{28}-a^{15}+a^{14}+a^{8}-a^{7}-1$, $2a^{33}-3a^{32}+3a^{31}-3a^{30}+3a^{29}-3a^{28}+3a^{27}-3a^{26}+3a^{25}-3a^{24}+3a^{23}-2a^{22}+2a^{21}-a^{20}+a^{19}-a^{18}-a^{16}+a^{12}-a^{11}+a^{10}-a^{9}+a^{8}-a^{7}+a^{6}+a^{4}-a^{3}+a^{2}-a-2$, $a^{32}-2a^{31}+a^{30}+a^{28}-a^{27}+a^{24}-a^{23}+a^{20}-a^{18}+a^{16}-a^{14}+a^{12}-a^{10}+a^{8}+a^{7}-a^{6}-a^{5}+a^{4}+a^{3}-a^{2}-a+1$, $a^{33}-a^{32}+a^{31}-a^{30}+a^{29}-a^{28}+a^{27}-a^{26}+a^{25}-a^{24}+a^{12}+a^{10}-a^{5}-1$, $a^{33}-a^{32}+a^{31}-a^{30}+a^{29}-a^{28}-a^{26}+a^{25}+a^{18}-a^{13}-a^{7}+a^{5}+a^{4}-1$, $2a^{33}-2a^{32}+2a^{31}-2a^{30}+a^{29}-a^{28}+2a^{27}-a^{26}+a^{25}-a^{24}+a^{23}+a^{19}+a^{14}-a^{9}+a^{6}-a^{4}-a^{3}-2$, $a^{33}-a^{27}+a^{26}+a^{12}-a^{11}+a^{6}-a^{2}-1$, $a^{33}-a^{32}-a^{30}-a^{27}+a^{26}+a^{23}+a^{20}-a^{18}+a^{17}-a^{16}-a^{14}-a^{12}-a^{9}+a^{8}+a^{7}+a^{5}+a^{4}+a^{2}-2$ (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: | \( 2995765078540.3994 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{16}\cdot 2995765078540.3994 \cdot 1}{2\cdot\sqrt{11885748945456377039253666531245020090157279092950049}}\cr\approx \mathstrut & 0.324266711279552 \end{aligned}\] (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 | $30{,}\,{\href{/padicField/2.4.0.1}{4} }$ | $17{,}\,{\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | $15{,}\,{\href{/padicField/11.9.0.1}{9} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }$ | $29{,}\,{\href{/padicField/17.5.0.1}{5} }$ | $31{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $31{,}\,{\href{/padicField/41.3.0.1}{3} }$ | $17{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | $28{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.15.0.1 | $x^{15} + 5 x^{6} + 6 x^{5} + 6 x^{4} + 4 x^{3} + x^{2} + 2 x + 4$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
7.15.0.1 | $x^{15} + 5 x^{6} + 6 x^{5} + 6 x^{4} + 4 x^{3} + x^{2} + 2 x + 4$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
\(191\) | 191.2.0.1 | $x^{2} + 190 x + 19$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
191.2.0.1 | $x^{2} + 190 x + 19$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
191.2.1.2 | $x^{2} + 191$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
191.11.0.1 | $x^{11} + 6 x + 172$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
191.17.0.1 | $x^{17} + 2 x + 172$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | |
\(607\) | $\Q_{607}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{607}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{607}$ | $x$ | $1$ | $1$ | $0$ | 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 $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(146\!\cdots\!111\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ |