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: | \(-113671059264051186521849878241745183709307592588026933074562263\) \(\medspace = -\,2273\cdot 50\!\cdots\!31\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(39.01\) | 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: | $2273^{1/2}50009264964386795654135450172347199168195157319853468136631^{1/2}\approx 1.0661663062770797e+31$ | ||
Ramified primes: | \(2273\), \(50009\!\cdots\!36631\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-11367\!\cdots\!62263}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a^{38}+1$, $a^{38}+a^{37}-a^{11}+1$, $a^{29}+a^{28}+a^{27}$, $a^{12}+a^{10}$, $a^{7}+a^{3}$, $a^{35}+a^{34}+a^{33}+a^{32}+a^{31}$, $a^{38}+a^{37}+a^{8}+1$, $a^{38}+a^{37}+a^{36}+a^{35}+a^{34}+a^{33}+a^{14}+1$, $a^{38}+a^{37}+a^{36}+a^{35}+a^{34}+a^{33}+a^{32}+a^{31}+a^{30}+a^{29}+a^{28}+a^{27}+a^{26}+a^{24}+a^{22}+1$, $a^{35}+a^{34}+a^{25}+a^{24}+a^{15}+a^{14}+a^{5}$, $a^{38}+a^{37}+a^{36}+a^{35}+a^{34}+a^{33}+a^{32}+a^{31}+a^{30}+a^{28}+a^{26}+a^{24}+a^{22}+a^{20}+1$, $a^{32}+a^{31}+a^{30}+a^{29}+a^{18}+a^{4}$, $a^{35}+a^{31}+a^{27}+a^{23}+a^{19}+a^{15}+a^{11}+a^{4}-a+1$, $a^{38}+a^{37}+a^{36}+a^{35}+a^{34}-a^{32}-a^{31}-a^{30}-a^{29}-2a^{28}-2a^{27}-2a^{26}-a^{25}-a^{24}-a^{23}-a^{22}+a^{20}+a^{19}+a^{18}+a^{16}+a^{15}+a^{14}-a^{8}-a^{7}-a^{6}-a^{3}-a^{2}+1$, $a^{38}+2a^{37}+2a^{36}+2a^{35}+3a^{34}+2a^{33}+2a^{32}+2a^{31}+2a^{30}+2a^{29}+2a^{28}+2a^{27}+2a^{26}+2a^{25}+a^{24}+a^{23}+a^{22}-a^{16}-a^{13}-a^{12}-a^{11}-a^{10}-a^{9}-a^{8}-a^{7}-a^{5}-a^{4}-a^{2}-a+1$, $a^{37}+2a^{36}+a^{35}+2a^{34}+a^{33}+a^{32}+a^{29}-a^{28}-a^{27}-a^{26}-a^{25}-a^{23}+a^{22}+a^{19}-a^{16}-2a^{13}-a^{11}+a^{7}-a^{6}+a^{5}+a^{4}-a+1$, $2a^{38}+4a^{37}+3a^{36}+3a^{35}+2a^{34}+2a^{33}+3a^{32}+a^{31}+2a^{30}+a^{29}+2a^{28}+2a^{27}+3a^{25}+a^{24}+2a^{23}+a^{22}+3a^{20}-a^{19}+a^{18}+a^{17}+2a^{15}-2a^{14}+2a^{13}+a^{12}-a^{11}+2a^{10}-a^{9}+2a^{8}-a^{6}+2a^{5}-a^{4}+a^{3}-a^{2}+4$, $a^{38}+a^{37}+a^{36}-a^{33}+a^{30}-a^{27}+a^{24}+a^{23}+a^{22}+a^{21}+a^{19}+a^{17}-a^{7}-a^{5}+a^{4}+a^{2}+1$, $a^{38}-a^{36}-2a^{35}-2a^{34}-3a^{33}-a^{32}-a^{31}+a^{29}+2a^{28}+3a^{27}+2a^{26}+2a^{25}-2a^{22}-3a^{21}-2a^{20}-2a^{19}-a^{18}-a^{17}+a^{16}+a^{15}+2a^{14}+2a^{13}+a^{12}+2a^{11}-a^{8}-a^{7}-2a^{6}-2a^{5}-a^{3}+a^{2}+2$ (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: | \( 1954406735071007.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{19}\cdot 1954406735071007.5 \cdot 1}{2\cdot\sqrt{113671059264051186521849878241745183709307592588026933074562263}}\cr\approx \mathstrut & 0.268291859705467 \end{aligned}\] (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 $S_{39}$ 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 | $26{,}\,{\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | $23{,}\,{\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $37{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | $39$ | $27{,}\,{\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $18{,}\,17{,}\,{\href{/padicField/17.4.0.1}{4} }$ | $35{,}\,{\href{/padicField/19.4.0.1}{4} }$ | $19{,}\,{\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | $22{,}\,{\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $26{,}\,{\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | $21{,}\,{\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $30{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $34{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $36{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2273\) | $\Q_{2273}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2273}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
\(500\!\cdots\!631\) | $\Q_{50\!\cdots\!31}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{50\!\cdots\!31}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{50\!\cdots\!31}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ |