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)
| |
Discriminant: | \(-18770034004737848533180532351835033630345588173537542144\) \(\medspace = -\,2^{30}\cdot 11\cdot 1489\cdot 1499\cdot 23334601\cdot 28850169461\cdot 1057614572100224186051\) | 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: | \(60.68\) | 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: | $2^{30/31}11^{1/2}1489^{1/2}1499^{1/2}23334601^{1/2}28850169461^{1/2}1057614572100224186051^{1/2}\approx 2.5858417634744722e+23$ | ||
Ramified primes: | \(2\), \(11\), \(1489\), \(1499\), \(23334601\), \(28850169461\), \(1057614572100224186051\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-17480\!\cdots\!34431}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a-1$, $a^{21}-a^{12}+a^{11}-a^{2}+1$, $a^{29}-a^{25}-a^{22}+a^{18}-a^{17}-2a^{16}+a^{13}+a^{12}-2a^{11}+2a^{9}-a^{8}-a^{7}+2a^{4}-a^{2}+2a-1$, $2a^{30}+a^{29}+a^{28}+2a^{27}+2a^{25}+a^{23}+a^{22}-a^{21}+2a^{20}-2a^{19}+2a^{18}-a^{17}+a^{15}-a^{14}+2a^{13}-a^{12}+a^{11}+a^{10}-a^{9}+3a^{8}-3a^{7}+3a^{6}-a^{5}+2a^{3}-3a^{2}+4a+1$, $a^{30}+2a^{29}+a^{28}-a^{25}-a^{24}+a^{22}+a^{20}-a^{18}-a^{17}-a^{15}+a^{14}+a^{13}-a^{11}-2a^{9}-a^{8}+a^{7}+a^{6}+a^{5}+2a^{4}-2a^{2}-a+1$, $a^{29}+a^{24}-a^{22}+a^{18}+a^{16}+a^{15}-2a^{14}+2a^{12}-a^{11}-a^{10}+a^{9}-a^{6}+a^{5}+3a^{4}-2a^{3}-2a^{2}+2a-1$, $a^{30}+a^{29}+a^{28}+a^{27}+a^{26}+2a^{25}+2a^{24}+a^{23}+a^{22}+a^{21}-a^{19}-a^{16}-a^{13}-2a^{12}-a^{11}-a^{9}+a^{7}-a^{5}-2a^{2}+3$, $a^{28}+a^{27}+2a^{26}+a^{25}+2a^{20}+a^{18}-a^{17}-a^{16}+a^{15}+2a^{13}-a^{10}+3a^{7}-3a^{3}+3a^{2}-2a+3$, $6a^{30}+3a^{29}-a^{28}-4a^{27}-6a^{26}-6a^{25}-4a^{24}-a^{23}+2a^{22}+4a^{21}+5a^{20}+4a^{19}+2a^{18}-a^{17}-3a^{16}-5a^{15}-5a^{14}-4a^{13}-a^{12}+a^{11}+4a^{10}+5a^{9}+5a^{8}+2a^{7}-4a^{5}-6a^{4}-7a^{3}-4a^{2}-2a+15$, $a^{30}+a^{29}-2a^{27}-3a^{26}-2a^{25}+a^{22}+2a^{21}+a^{20}-2a^{19}-3a^{18}-2a^{17}-a^{16}+a^{15}+2a^{14}+2a^{13}-2a^{10}-4a^{9}-2a^{8}+2a^{7}+3a^{6}+a^{5}+a^{4}-2a^{2}-4a+1$, $a^{29}-a^{28}-2a^{27}+2a^{23}-2a^{21}+3a^{17}+a^{16}-2a^{15}-a^{13}+3a^{11}+a^{10}-a^{9}-a^{8}-2a^{7}+a^{6}+a^{5}+2a^{4}+a^{3}-3a^{2}-a+1$, $2a^{30}+3a^{29}+2a^{28}+a^{27}-a^{25}-a^{24}-a^{23}-2a^{22}-a^{21}+a^{20}+a^{19}+2a^{18}+3a^{17}+2a^{16}-a^{14}-2a^{13}-3a^{12}-a^{9}+2a^{8}+2a^{7}+a^{6}+2a^{5}+2a^{4}-a^{3}-3a^{2}-a+1$, $a^{30}-a^{29}-2a^{28}-a^{27}-2a^{25}-4a^{24}-3a^{23}-a^{22}-2a^{21}-4a^{20}-4a^{19}-2a^{18}-2a^{16}-4a^{15}-2a^{14}+a^{13}+a^{12}-2a^{11}-2a^{10}+a^{9}+3a^{8}+a^{7}-2a^{6}+4a^{4}+4a^{3}-a^{2}-2a+5$, $5a^{30}+6a^{29}+5a^{28}+6a^{27}+4a^{26}+5a^{25}+4a^{24}+4a^{23}+4a^{22}+3a^{21}+2a^{20}+a^{19}+a^{18}-a^{17}-a^{16}-2a^{14}-a^{13}-3a^{12}-a^{11}-5a^{10}-a^{9}-5a^{8}-a^{7}-3a^{6}-a^{5}-3a^{4}-a^{3}-a^{2}-4a+11$, $a^{30}-a^{29}+a^{28}-2a^{27}-a^{26}+a^{25}-3a^{24}-a^{22}-2a^{21}+2a^{20}-2a^{19}+2a^{18}+a^{17}-2a^{16}+4a^{15}-a^{14}+3a^{12}-2a^{11}+2a^{10}-2a^{9}-3a^{8}+4a^{7}-5a^{6}+a^{5}+a^{4}-5a^{3}+5a^{2}-3a+3$ (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)]
| |
Regulator: | \( 5116384966428125.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{15}\cdot 5116384966428125.0 \cdot 1}{2\cdot\sqrt{18770034004737848533180532351835033630345588173537542144}}\cr\approx \mathstrut & 1.10899275730712 \end{aligned}\] (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 $S_{31}$ 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 | R | $29{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | $16{,}\,{\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $24{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | $20{,}\,{\href{/padicField/29.11.0.1}{11} }$ | ${\href{/padicField/31.10.0.1}{10} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.12.0.1}{12} }^{2}{,}\,{\href{/padicField/37.7.0.1}{7} }$ | $18{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | $28{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 $31$ | $31$ | $1$ | $30$ | |||
\(11\) | 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
11.12.0.1 | $x^{12} + x^{8} + x^{7} + 4 x^{6} + 2 x^{5} + 5 x^{4} + 5 x^{3} + 6 x^{2} + 5 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
11.14.0.1 | $x^{14} + 2 x^{7} + 9 x^{6} + 6 x^{5} + 4 x^{4} + 8 x^{3} + 6 x^{2} + 10 x + 2$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
\(1489\) | $\Q_{1489}$ | $x$ | $1$ | $1$ | $0$ | 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 $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(1499\) | $\Q_{1499}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1499}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $27$ | $1$ | $27$ | $0$ | $C_{27}$ | $[\ ]^{27}$ | ||
\(23334601\) | $\Q_{23334601}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{23334601}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{23334601}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | ||
\(28850169461\) | $\Q_{28850169461}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(105\!\cdots\!051\) | $\Q_{10\!\cdots\!51}$ | $x$ | $1$ | $1$ | $0$ | 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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |