Normalized defining polynomial
\( x^{31} + 2x - 1 \)
Invariants
Degree: | $31$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-3654114516616866500188330236451595281481940132601111\) \(\medspace = -\,71\cdot 6073\cdot 84\!\cdots\!17\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(46.06\) | 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: | $71^{1/2}6073^{1/2}8474625661533192403662320259499088047260537017^{1/2}\approx 6.044927225878627e+25$ | ||
Ramified primes: | \(71\), \(6073\), \(84746\!\cdots\!37017\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-36541\!\cdots\!01111}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{11}a^{30}+\frac{4}{11}a^{29}+\frac{5}{11}a^{28}-\frac{2}{11}a^{27}+\frac{3}{11}a^{26}+\frac{1}{11}a^{25}+\frac{4}{11}a^{24}+\frac{5}{11}a^{23}-\frac{2}{11}a^{22}+\frac{3}{11}a^{21}+\frac{1}{11}a^{20}+\frac{4}{11}a^{19}+\frac{5}{11}a^{18}-\frac{2}{11}a^{17}+\frac{3}{11}a^{16}+\frac{1}{11}a^{15}+\frac{4}{11}a^{14}+\frac{5}{11}a^{13}-\frac{2}{11}a^{12}+\frac{3}{11}a^{11}+\frac{1}{11}a^{10}+\frac{4}{11}a^{9}+\frac{5}{11}a^{8}-\frac{2}{11}a^{7}+\frac{3}{11}a^{6}+\frac{1}{11}a^{5}+\frac{4}{11}a^{4}+\frac{5}{11}a^{3}-\frac{2}{11}a^{2}+\frac{3}{11}a+\frac{3}{11}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $15$ | 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: | $a^{30}+2$, $a^{4}-a^{2}+1$, $a^{20}-a^{10}+1$, $a^{24}-a^{18}+a^{12}-a^{6}+1$, $\frac{5}{11}a^{30}-\frac{2}{11}a^{29}+\frac{3}{11}a^{28}+\frac{1}{11}a^{27}+\frac{4}{11}a^{26}+\frac{5}{11}a^{25}-\frac{2}{11}a^{24}+\frac{3}{11}a^{23}+\frac{1}{11}a^{22}+\frac{4}{11}a^{21}+\frac{5}{11}a^{20}-\frac{2}{11}a^{19}+\frac{3}{11}a^{18}+\frac{1}{11}a^{17}+\frac{4}{11}a^{16}+\frac{5}{11}a^{15}-\frac{2}{11}a^{14}-\frac{8}{11}a^{13}+\frac{1}{11}a^{12}-\frac{7}{11}a^{11}+\frac{5}{11}a^{10}-\frac{2}{11}a^{9}-\frac{8}{11}a^{8}+\frac{1}{11}a^{7}-\frac{7}{11}a^{6}+\frac{5}{11}a^{5}-\frac{2}{11}a^{4}-\frac{8}{11}a^{3}+\frac{1}{11}a^{2}-\frac{7}{11}a+\frac{15}{11}$, $\frac{6}{11}a^{30}+\frac{13}{11}a^{29}+\frac{19}{11}a^{28}+\frac{10}{11}a^{27}-\frac{4}{11}a^{26}-\frac{5}{11}a^{25}+\frac{2}{11}a^{24}+\frac{8}{11}a^{23}+\frac{10}{11}a^{22}+\frac{7}{11}a^{21}+\frac{6}{11}a^{20}+\frac{2}{11}a^{19}-\frac{3}{11}a^{18}-\frac{1}{11}a^{17}+\frac{7}{11}a^{16}+\frac{17}{11}a^{15}+\frac{13}{11}a^{14}-\frac{3}{11}a^{13}-\frac{12}{11}a^{12}-\frac{4}{11}a^{11}+\frac{6}{11}a^{10}+\frac{13}{11}a^{9}+\frac{8}{11}a^{8}-\frac{1}{11}a^{7}-\frac{4}{11}a^{6}-\frac{5}{11}a^{5}-\frac{9}{11}a^{4}-\frac{3}{11}a^{3}+\frac{10}{11}a^{2}+\frac{7}{11}a+\frac{7}{11}$, $\frac{1}{11}a^{30}+\frac{4}{11}a^{29}+\frac{5}{11}a^{28}+\frac{9}{11}a^{27}+\frac{3}{11}a^{26}+\frac{1}{11}a^{25}+\frac{4}{11}a^{24}-\frac{6}{11}a^{23}-\frac{2}{11}a^{22}-\frac{8}{11}a^{21}+\frac{1}{11}a^{20}-\frac{7}{11}a^{19}+\frac{5}{11}a^{18}-\frac{2}{11}a^{17}+\frac{14}{11}a^{16}-\frac{10}{11}a^{15}+\frac{15}{11}a^{14}-\frac{6}{11}a^{13}+\frac{9}{11}a^{12}-\frac{19}{11}a^{11}+\frac{12}{11}a^{10}-\frac{18}{11}a^{9}+\frac{5}{11}a^{8}-\frac{13}{11}a^{7}+\frac{14}{11}a^{6}-\frac{10}{11}a^{5}+\frac{15}{11}a^{4}-\frac{6}{11}a^{3}+\frac{20}{11}a^{2}-\frac{19}{11}a+\frac{14}{11}$, $\frac{5}{11}a^{30}+\frac{9}{11}a^{29}+\frac{14}{11}a^{28}+\frac{12}{11}a^{27}+\frac{4}{11}a^{26}-\frac{6}{11}a^{25}-\frac{13}{11}a^{24}-\frac{8}{11}a^{23}+\frac{1}{11}a^{22}+\frac{4}{11}a^{21}+\frac{5}{11}a^{20}+\frac{9}{11}a^{19}+\frac{14}{11}a^{18}+\frac{1}{11}a^{17}-\frac{18}{11}a^{16}-\frac{17}{11}a^{15}-\frac{2}{11}a^{14}+\frac{3}{11}a^{13}+\frac{1}{11}a^{12}+\frac{4}{11}a^{11}+\frac{16}{11}a^{10}+\frac{9}{11}a^{9}-\frac{8}{11}a^{8}-\frac{21}{11}a^{7}-\frac{18}{11}a^{6}-\frac{6}{11}a^{5}+\frac{9}{11}a^{4}+\frac{14}{11}a^{3}+\frac{12}{11}a^{2}+\frac{4}{11}a+\frac{4}{11}$, $\frac{7}{11}a^{30}+\frac{17}{11}a^{29}+\frac{2}{11}a^{28}-\frac{14}{11}a^{27}-\frac{23}{11}a^{26}-\frac{15}{11}a^{25}+\frac{6}{11}a^{24}+\frac{13}{11}a^{23}+\frac{19}{11}a^{22}+\frac{10}{11}a^{21}-\frac{4}{11}a^{20}-\frac{16}{11}a^{19}-\frac{20}{11}a^{18}-\frac{3}{11}a^{17}+\frac{10}{11}a^{16}+\frac{18}{11}a^{15}+\frac{6}{11}a^{14}-\frac{9}{11}a^{13}-\frac{3}{11}a^{12}-\frac{1}{11}a^{11}+\frac{7}{11}a^{10}-\frac{5}{11}a^{9}-\frac{9}{11}a^{8}-\frac{3}{11}a^{7}-\frac{1}{11}a^{6}+\frac{7}{11}a^{5}+\frac{6}{11}a^{4}+\frac{13}{11}a^{3}+\frac{8}{11}a^{2}-\frac{12}{11}a-\frac{12}{11}$, $\frac{5}{11}a^{30}+\frac{9}{11}a^{29}-\frac{8}{11}a^{28}+\frac{1}{11}a^{27}+\frac{4}{11}a^{26}+\frac{5}{11}a^{25}+\frac{9}{11}a^{24}-\frac{8}{11}a^{23}+\frac{1}{11}a^{22}-\frac{7}{11}a^{21}+\frac{5}{11}a^{20}+\frac{9}{11}a^{19}-\frac{8}{11}a^{18}+\frac{1}{11}a^{17}-\frac{7}{11}a^{16}+\frac{16}{11}a^{15}-\frac{2}{11}a^{14}-\frac{8}{11}a^{13}+\frac{1}{11}a^{12}-\frac{18}{11}a^{11}+\frac{16}{11}a^{10}-\frac{2}{11}a^{9}+\frac{3}{11}a^{8}+\frac{1}{11}a^{7}-\frac{18}{11}a^{6}+\frac{16}{11}a^{5}-\frac{13}{11}a^{4}+\frac{3}{11}a^{3}+\frac{1}{11}a^{2}-\frac{7}{11}a+\frac{15}{11}$, $\frac{17}{11}a^{30}+\frac{13}{11}a^{29}+\frac{8}{11}a^{28}+\frac{10}{11}a^{27}-\frac{4}{11}a^{26}+\frac{6}{11}a^{25}+\frac{2}{11}a^{24}+\frac{8}{11}a^{23}+\frac{10}{11}a^{22}+\frac{7}{11}a^{21}+\frac{6}{11}a^{20}-\frac{9}{11}a^{19}+\frac{8}{11}a^{18}-\frac{1}{11}a^{17}+\frac{7}{11}a^{16}+\frac{17}{11}a^{15}-\frac{9}{11}a^{14}+\frac{8}{11}a^{13}-\frac{12}{11}a^{12}-\frac{4}{11}a^{11}+\frac{6}{11}a^{10}+\frac{2}{11}a^{9}+\frac{8}{11}a^{8}-\frac{12}{11}a^{7}-\frac{4}{11}a^{6}-\frac{16}{11}a^{5}+\frac{2}{11}a^{4}+\frac{8}{11}a^{3}-\frac{12}{11}a^{2}+\frac{18}{11}a+\frac{7}{11}$, $\frac{8}{11}a^{30}-\frac{1}{11}a^{29}-\frac{4}{11}a^{28}-\frac{5}{11}a^{27}-\frac{9}{11}a^{26}+\frac{8}{11}a^{25}-\frac{1}{11}a^{24}-\frac{4}{11}a^{23}+\frac{17}{11}a^{22}+\frac{2}{11}a^{21}-\frac{3}{11}a^{20}-\frac{1}{11}a^{19}-\frac{15}{11}a^{18}-\frac{5}{11}a^{17}+\frac{2}{11}a^{16}-\frac{3}{11}a^{15}+\frac{10}{11}a^{14}+\frac{7}{11}a^{13}-\frac{5}{11}a^{12}+\frac{2}{11}a^{11}-\frac{3}{11}a^{10}-\frac{12}{11}a^{9}-\frac{4}{11}a^{8}+\frac{6}{11}a^{7}+\frac{13}{11}a^{6}+\frac{8}{11}a^{5}-\frac{1}{11}a^{4}-\frac{4}{11}a^{3}+\frac{6}{11}a^{2}-\frac{9}{11}a-\frac{9}{11}$, $\frac{6}{11}a^{30}+\frac{2}{11}a^{29}-\frac{3}{11}a^{28}-\frac{1}{11}a^{27}+\frac{7}{11}a^{26}+\frac{6}{11}a^{25}+\frac{2}{11}a^{24}-\frac{3}{11}a^{23}-\frac{1}{11}a^{22}+\frac{7}{11}a^{21}+\frac{6}{11}a^{20}+\frac{2}{11}a^{19}-\frac{14}{11}a^{18}-\frac{12}{11}a^{17}-\frac{4}{11}a^{16}+\frac{6}{11}a^{15}+\frac{2}{11}a^{14}-\frac{3}{11}a^{13}-\frac{1}{11}a^{12}-\frac{4}{11}a^{11}+\frac{6}{11}a^{10}+\frac{13}{11}a^{9}-\frac{3}{11}a^{8}-\frac{1}{11}a^{7}-\frac{4}{11}a^{6}+\frac{6}{11}a^{5}+\frac{13}{11}a^{4}+\frac{8}{11}a^{3}-\frac{12}{11}a^{2}-\frac{15}{11}a+\frac{18}{11}$, $\frac{12}{11}a^{30}-\frac{18}{11}a^{29}+\frac{5}{11}a^{28}-\frac{2}{11}a^{27}-\frac{8}{11}a^{26}+\frac{12}{11}a^{25}-\frac{18}{11}a^{24}+\frac{16}{11}a^{23}-\frac{13}{11}a^{22}-\frac{8}{11}a^{21}+\frac{12}{11}a^{20}-\frac{18}{11}a^{19}+\frac{16}{11}a^{18}-\frac{24}{11}a^{17}+\frac{14}{11}a^{16}+\frac{1}{11}a^{15}-\frac{18}{11}a^{14}+\frac{27}{11}a^{13}-\frac{24}{11}a^{12}+\frac{25}{11}a^{11}-\frac{10}{11}a^{10}+\frac{4}{11}a^{9}+\frac{16}{11}a^{8}-\frac{35}{11}a^{7}+\frac{47}{11}a^{6}-\frac{32}{11}a^{5}+\frac{15}{11}a^{4}-\frac{6}{11}a^{3}-\frac{13}{11}a^{2}+\frac{36}{11}a-\frac{30}{11}$, $\frac{9}{11}a^{30}-\frac{30}{11}a^{29}+\frac{12}{11}a^{28}+\frac{4}{11}a^{27}-\frac{6}{11}a^{26}+\frac{31}{11}a^{25}-\frac{19}{11}a^{24}-\frac{32}{11}a^{23}+\frac{26}{11}a^{22}-\frac{6}{11}a^{21}+\frac{9}{11}a^{20}+\frac{14}{11}a^{19}-\frac{43}{11}a^{18}-\frac{7}{11}a^{17}+\frac{27}{11}a^{16}-\frac{13}{11}a^{15}+\frac{14}{11}a^{14}-\frac{10}{11}a^{13}-\frac{51}{11}a^{12}+\frac{38}{11}a^{11}+\frac{9}{11}a^{10}-\frac{19}{11}a^{9}+\frac{23}{11}a^{8}-\frac{40}{11}a^{7}-\frac{28}{11}a^{6}+\frac{64}{11}a^{5}-\frac{19}{11}a^{4}-\frac{21}{11}a^{3}+\frac{37}{11}a^{2}-\frac{72}{11}a+\frac{38}{11}$ (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: | \( 38986878555233.18 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{15}\cdot 38986878555233.18 \cdot 1}{2\cdot\sqrt{3654114516616866500188330236451595281481940132601111}}\cr\approx \mathstrut & 0.605655149069965 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 8222838654177922817725562880000000 |
The 6842 conjugacy class representatives for $S_{31}$ are not computed |
Character table for $S_{31}$ |
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 | ${\href{/padicField/2.5.0.1}{5} }^{6}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $31$ | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $18{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | $17{,}\,{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.10.0.1}{10} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $26{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $19{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $23{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(71\) | $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.5.0.1 | $x^{5} + 18 x + 64$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
71.23.0.1 | $x^{23} + 4 x + 64$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | |
\(6073\) | $\Q_{6073}$ | $x$ | $1$ | $1$ | $0$ | 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 $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ | ||
\(847\!\cdots\!017\) | $\Q_{84\!\cdots\!17}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ |