Normalized defining polynomial
\( x^{31} - 2 \)
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)
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Discriminant: | \(-18327886165296381817380980351835033630345588173537542144\) \(\medspace = -\,2^{30}\cdot 31^{31}\) | 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: | \(60.63\) | 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: | $2^{30/31}31^{959/930}\approx 67.48167773073806$ | ||
Ramified primes: | \(2\), \(31\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-31}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$
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)
| |
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: | $a-1$, $a^{16}-a-1$, $a^{17}+a^{16}-a^{2}-a-1$, $a^{29}+a^{27}+a^{25}+a^{23}+a^{21}+a^{19}+a^{17}+a^{15}+a^{13}+a^{11}+a^{9}+a^{7}+a^{5}+a^{3}+a+1$, $a^{29}-a^{27}+a^{25}-a^{23}+a^{21}-a^{19}-a^{14}+a^{12}-a^{10}+a^{8}-a^{6}+a^{4}+a-1$, $a^{29}+a^{26}-a^{25}-a^{22}-a^{16}-a^{14}-a^{11}+a^{10}+a^{7}-a^{5}+2a^{4}+2a-1$, $a^{12}+a^{11}+a^{10}+a^{9}+a^{8}+a^{7}-a^{5}-a^{4}-a^{3}-a^{2}-2a-1$, $a^{30}+a^{28}-a^{25}+a^{22}-a^{19}-a^{18}+a^{16}+a^{15}+a^{14}+a^{13}-a^{12}-a^{11}-2a^{10}-a^{9}+2a^{7}+2a^{6}+2a^{5}-a^{4}-a^{3}-2a^{2}-a-1$, $a^{30}-a^{28}-a^{27}-a^{26}+a^{25}-a^{22}+a^{21}+a^{20}+2a^{19}+2a^{14}-2a^{13}-a^{12}-2a^{11}-a^{7}-a^{6}+a^{5}+2a^{4}+2a^{3}-2a^{2}+a-1$, $a^{30}+a^{29}+2a^{24}+3a^{23}+a^{22}+a^{20}-a^{18}+a^{16}+a^{15}+a^{14}-a^{13}-3a^{12}-2a^{11}-a^{10}-a^{9}-a^{8}-a^{5}-3a^{4}-3a^{3}+a+1$, $a^{30}-a^{29}-a^{28}-3a^{26}-2a^{25}+4a^{24}+2a^{23}+2a^{21}+a^{20}-2a^{19}-a^{18}-4a^{17}-5a^{16}+2a^{15}+2a^{14}-a^{13}+3a^{12}+6a^{11}+a^{10}+a^{9}-a^{8}-5a^{7}-a^{6}-5a^{4}-a^{3}+7a^{2}+2a+1$, $4a^{30}-3a^{28}-3a^{27}-3a^{26}+a^{25}+4a^{24}+4a^{23}+3a^{22}-5a^{20}-4a^{19}-3a^{18}+5a^{16}+4a^{15}+a^{14}-2a^{13}-6a^{12}-6a^{11}-a^{10}+a^{9}+6a^{8}+6a^{7}+a^{6}-2a^{5}-6a^{4}-7a^{3}+5a+7$, $a^{29}-a^{28}-a^{25}+2a^{24}-a^{23}-2a^{20}+3a^{19}-a^{18}-2a^{15}+3a^{14}-a^{13}-2a^{10}+2a^{9}-2a^{5}+a^{4}+a^{3}-a^{2}-1$, $a^{28}+a^{27}-a^{24}-2a^{23}-a^{22}+a^{21}+a^{20}+a^{19}+a^{18}+2a^{17}+2a^{16}+a^{15}-3a^{14}-4a^{13}-2a^{12}-a^{10}+a^{8}+3a^{7}+4a^{6}+2a^{5}-2a^{4}-a^{3}-a^{2}-2a-1$, $2a^{30}-2a^{29}+a^{27}+2a^{25}-2a^{24}+2a^{22}-a^{21}+a^{20}-2a^{19}+3a^{17}-2a^{16}+a^{15}+3a^{12}-4a^{11}-a^{10}+a^{9}-2a^{8}+2a^{7}-3a^{6}+4a^{4}-3a^{3}-2a-1$ (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: | \( 5190129910021260.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{15}\cdot 5190129910021260.0 \cdot 1}{2\cdot\sqrt{18327886165296381817380980351835033630345588173537542144}}\cr\approx \mathstrut & 1.13846599841673 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 930 |
The 31 conjugacy class representatives for $F_{31}$ |
Character table for $F_{31}$ |
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 | $30{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.3.0.1}{3} }^{10}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $15^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $30{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $30{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $30{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $15^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.10.0.1}{10} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.10.0.1}{10} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | R | ${\href{/padicField/37.6.0.1}{6} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $15^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $30{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.5.0.1}{5} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $30{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $15^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $31$ | $31$ | $1$ | $30$ | |||
\(31\) | Deg $31$ | $31$ | $1$ | $31$ |