Normalized defining polynomial
\( x^{31} - 2x - 2 \)
Invariants
Degree: | $31$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-17885738325854915101581428351835033630345588173537542144\) \(\medspace = -\,2^{30}\cdot 937\cdot 17\!\cdots\!63\) | 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: | \(60.58\) | 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))
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Galois root discriminant: | $2^{30/31}937^{1/2}17777365919459912442487345815431499206024263^{1/2}\approx 2.5241947084706398e+23$ | ||
Ramified primes: | \(2\), \(937\), \(17777\!\cdots\!24263\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | $\Q(\sqrt{-16657\!\cdots\!34431}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$
Monogenic: | Yes | |
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)
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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)
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Fundamental units: | $a+1$, $a^{21}+a^{11}+a+1$, $a^{16}-a-1$, $a^{30}+a^{28}+a^{26}+a^{24}+a^{22}+a^{20}+a^{18}-a^{15}-a^{14}-a^{13}-a^{12}-a^{11}-a^{10}-a^{9}-a^{8}-a^{7}-a^{6}-a^{5}-a^{4}-a^{3}-a^{2}-1$, $a^{28}-a^{26}+a^{23}-a^{21}+a^{18}-2a^{15}+a^{14}+a^{13}-2a^{10}+a^{9}+a^{8}+a^{7}-a^{6}-2a^{5}+2a^{3}+a^{2}-2a-1$, $5a^{30}-4a^{29}+4a^{28}-3a^{27}+3a^{26}-3a^{25}+2a^{24}-2a^{23}+a^{22}-a^{21}+a^{20}-a^{19}+a^{15}-a^{12}+a^{11}-a^{10}+a^{9}-a^{8}-2a^{6}+a^{5}-a^{4}+2a^{3}+a-11$, $a^{30}+a^{29}+a^{28}+a^{27}-a^{24}-a^{23}-a^{22}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}-a^{11}-2a^{10}-4a^{9}-4a^{8}-4a^{7}-3a^{6}-a^{5}+a^{3}+2a^{2}+2a+1$, $a^{26}-2a^{25}+a^{24}-a^{17}+a^{16}-2a^{15}+a^{11}+a^{8}+a^{6}-2a^{5}+a^{3}-a^{2}+a+1$, $a^{29}+a^{28}+a^{25}+2a^{24}+a^{23}-a^{22}-a^{21}+a^{19}+a^{18}+a^{15}+2a^{14}+2a^{13}-a^{12}-2a^{11}-a^{10}+2a^{9}+2a^{8}+2a^{5}+4a^{4}+3a^{3}-2a-1$, $2a^{29}+a^{27}-2a^{24}-2a^{22}+a^{20}+a^{19}+a^{18}+2a^{17}+a^{16}-2a^{15}-3a^{13}-a^{12}-a^{11}+a^{10}+4a^{8}+2a^{7}-2a^{4}-3a^{3}-2a^{2}-a-1$, $2a^{30}-a^{29}+a^{27}+a^{23}-a^{21}+2a^{20}-a^{19}+a^{16}-a^{15}-a^{14}+2a^{13}-2a^{12}-a^{8}+a^{6}-2a^{5}+a^{4}+a^{3}-a^{2}-3$, $2a^{27}-a^{26}+2a^{25}-2a^{24}+a^{23}-3a^{22}+a^{21}-2a^{20}+a^{19}+2a^{17}+a^{16}+a^{15}+a^{14}-2a^{13}-a^{12}-3a^{11}-2a^{9}+2a^{8}+a^{7}+2a^{6}+a^{5}+a^{4}-3a^{2}-a-3$, $3a^{30}-a^{29}-2a^{28}+3a^{27}-4a^{26}+5a^{25}-3a^{24}+2a^{23}-3a^{21}+3a^{20}-5a^{19}+5a^{18}-2a^{17}+2a^{16}+a^{15}-3a^{14}+3a^{13}-6a^{12}+4a^{11}-2a^{10}+2a^{9}+2a^{8}-3a^{7}+4a^{6}-6a^{5}+3a^{4}-2a^{3}+2a^{2}+2a-9$, $2a^{30}-2a^{28}-a^{27}+a^{26}+2a^{25}-3a^{23}-a^{22}+a^{21}+3a^{20}+a^{19}-3a^{18}-2a^{17}+a^{16}+4a^{15}+a^{14}-3a^{13}-3a^{12}+a^{11}+3a^{10}+2a^{9}-4a^{8}-4a^{7}+a^{6}+5a^{5}+3a^{4}-4a^{3}-4a^{2}+a+1$, $2a^{30}+a^{28}+a^{26}+2a^{24}-a^{23}+2a^{22}+a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+2a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+2a^{9}+a^{8}+2a^{7}+a^{6}+a^{5}+2a^{4}+2a^{3}+2a^{2}+a-3$ (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)]
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Regulator: | \( 3069030481487575.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{15}\cdot 3069030481487575.5 \cdot 1}{2\cdot\sqrt{17885738325854915101581428351835033630345588173537542144}}\cr\approx \mathstrut & 0.681468507829213 \end{aligned}\] (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 | R | $16{,}\,{\href{/padicField/3.11.0.1}{11} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $31$ | $30{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $31$ | $20{,}\,{\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | $25{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $26{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.10.0.1}{10} }^{2}{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | $26{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $31$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 $31$ | $31$ | $1$ | $30$ | |||
\(937\) | $\Q_{937}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{937}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{937}$ | $x$ | $1$ | $1$ | $0$ | 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 $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | ||
\(177\!\cdots\!263\) | $\Q_{17\!\cdots\!63}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{17\!\cdots\!63}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{17\!\cdots\!63}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ |