Normalized defining polynomial
\( x^{39} - x - 1 \)
Invariants
Degree: | $39$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 19]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-111519235660091198557729019736032936151076617850366100945341655\) \(\medspace = -\,5\cdot 277\cdot 182969\cdot 824281\cdot 240883723\cdot 166682962769\cdot 13296842994412570838812157821\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(38.99\) | 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: | $5^{1/2}277^{1/2}182969^{1/2}824281^{1/2}240883723^{1/2}166682962769^{1/2}13296842994412570838812157821^{1/2}\approx 1.0560266836595144e+31$ | ||
Ramified primes: | \(5\), \(277\), \(182969\), \(824281\), \(240883723\), \(166682962769\), \(13296842994412570838812157821\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-11151\!\cdots\!41655}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $19$ | 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: | $a^{38}-1$, $a^{20}-a$, $a^{14}-a$, $a^{27}-a^{26}$, $a^{38}-a^{37}+a^{36}-a^{35}+a^{34}-a^{33}+a^{32}-a^{31}+a^{30}-a^{29}+a^{28}-a^{27}+a^{26}-a^{25}+a^{24}+a^{22}-1$, $a^{38}-a^{37}-a^{6}-1$, $a^{11}-1$, $a^{38}-a^{37}+a^{36}-a^{35}+a^{34}-a^{33}+a^{30}-a^{29}-1$, $a^{11}-a^{4}$, $a^{11}-a^{8}+a^{5}$, $a^{38}-a^{37}+a^{36}-a^{35}+a^{34}-a^{33}+a^{32}-a^{31}+a^{30}-a^{29}+a^{28}-a^{27}+a^{26}-a^{25}+a^{24}-a^{23}+a^{22}-a^{21}-a^{19}+a^{18}+a^{16}-a^{13}-1$, $2a^{36}-a^{35}-a^{33}+a^{32}-a^{26}+a^{23}-a^{21}+a^{18}-a^{17}+a^{16}-a^{13}+a^{11}-a^{8}+a^{2}-a-1$, $a^{35}+a^{33}-a^{29}-a^{27}+a^{22}+a^{20}-a^{16}-a^{15}-a^{14}-a^{13}+a^{10}+a^{9}+a^{8}+a^{7}-a^{4}-a^{3}-a^{2}-a-1$, $a^{38}-a^{37}+a^{36}-a^{35}+a^{34}-a^{33}-a^{28}+a^{26}-a^{25}-a^{18}+a^{17}+a^{12}-a^{4}+a^{2}-a-1$, $2a^{38}-a^{37}+2a^{35}-3a^{34}+2a^{33}-a^{32}+a^{31}-2a^{29}+3a^{28}-2a^{27}+2a^{26}-2a^{25}+2a^{23}-2a^{22}+a^{21}-a^{20}+a^{19}+a^{18}-2a^{17}+2a^{16}-2a^{15}+2a^{14}-a^{13}-a^{12}+2a^{11}-a^{10}+a^{9}-a^{8}+a^{6}-a^{5}+a^{4}-2a^{3}+2a^{2}-3$, $a^{36}-a^{32}+a^{28}-a^{25}+a^{21}-a^{17}+a^{13}-a^{9}+a^{5}-a$, $a^{38}-a^{32}-a^{30}+a^{29}+a^{25}-a^{24}-a^{23}+a^{20}+a^{18}+a^{16}-a^{15}-a^{13}+a^{12}-a^{11}+a^{8}-a^{5}-a^{2}$, $3a^{38}-3a^{37}+3a^{36}-2a^{35}+2a^{34}-3a^{33}+3a^{32}-2a^{31}+2a^{30}-3a^{29}+2a^{28}-2a^{27}+2a^{26}-2a^{25}+3a^{24}-a^{23}+a^{22}-2a^{21}+2a^{20}-a^{19}+a^{18}-a^{17}+a^{16}-a^{15}-a^{13}+2a^{12}+a^{8}-a^{7}-a^{3}-a^{2}-2$, $a^{38}-2a^{37}+a^{36}+2a^{34}+2a^{33}+a^{32}+3a^{31}+2a^{30}+2a^{29}+a^{28}-a^{25}-2a^{24}-3a^{23}-2a^{22}-2a^{21}-3a^{20}-a^{19}-2a^{18}+a^{17}+2a^{15}+2a^{14}+4a^{13}+3a^{12}+2a^{11}+3a^{10}+a^{9}+2a^{8}-a^{7}-a^{6}-3a^{5}-2a^{4}-4a^{3}-4a^{2}-3a-4$ (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: | \( 2439954038147028.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{19}\cdot 2439954038147028.0 \cdot 1}{2\cdot\sqrt{111519235660091198557729019736032936151076617850366100945341655}}\cr\approx \mathstrut & 0.338161570864165 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 20397882081197443358640281739902897356800000000 |
The 31185 conjugacy class representatives for $S_{39}$ are not computed |
Character table for $S_{39}$ |
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 | $26{,}\,{\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | $27{,}\,{\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.11.0.1}{11} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $16^{2}{,}\,{\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.13.0.1}{13} }{,}\,{\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | $23{,}\,{\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.5.0.1}{5} }$ | $27{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $24{,}\,{\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $23{,}\,{\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.6.0.1}{6} }$ | $20{,}\,16{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.13.0.1}{13} }^{2}{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ | $20{,}\,{\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
5.13.0.1 | $x^{13} + 4 x^{2} + 3 x + 3$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
5.21.0.1 | $x^{21} + 4 x^{7} + 2 x^{6} + 2 x^{5} + 2 x^{4} + x^{3} + 2 x^{2} + 2 x + 3$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | |
\(277\) | $\Q_{277}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{277}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{277}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{277}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(182969\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(824281\) | $\Q_{824281}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{824281}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{824281}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
\(240883723\) | $\Q_{240883723}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(166682962769\) | $\Q_{166682962769}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
\(132\!\cdots\!821\) | $\Q_{13\!\cdots\!21}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ |