Normalized defining polynomial
\( x^{23} + 7x - 4 \)
Invariants
Degree: | $23$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-2227978859786797789863805466564380466715471\) \(\medspace = -\,3\cdot 661\cdot 102898348801\cdot 10918926580157025647925153137\) | 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: | \(69.38\) | 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: | $3^{1/2}661^{1/2}102898348801^{1/2}10918926580157025647925153137^{1/2}\approx 1.4926415711036585e+21$ | ||
Ramified primes: | \(3\), \(661\), \(102898348801\), \(10918926580157025647925153137\) | 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: | $\Q(\sqrt{-22279\!\cdots\!15471}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{9}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{10}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{11}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | 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: | $a^{22}+2a^{21}+2a^{20}+a^{19}-a^{18}-\frac{5}{2}a^{17}-3a^{16}-\frac{3}{2}a^{15}+\frac{1}{2}a^{14}+3a^{13}+\frac{7}{2}a^{12}+3a^{11}-3a^{9}-6a^{8}-4a^{7}-\frac{5}{2}a^{6}+5a^{5}+\frac{11}{2}a^{4}+\frac{15}{2}a^{3}+2a^{2}-\frac{5}{2}a-1$, $\frac{1}{2}a^{22}-a^{20}+\frac{1}{2}a^{19}+a^{18}-\frac{3}{2}a^{17}+\frac{1}{2}a^{16}+2a^{15}-\frac{5}{2}a^{14}+2a^{12}-\frac{3}{2}a^{11}-2a^{10}+4a^{9}-\frac{3}{2}a^{8}-a^{7}+\frac{3}{2}a^{6}+\frac{1}{2}a^{5}-2a^{4}-\frac{1}{2}a^{3}+a^{2}+2a-1$, $a^{21}-\frac{1}{2}a^{20}+\frac{3}{2}a^{19}+\frac{1}{2}a^{18}+\frac{1}{2}a^{17}+2a^{16}-\frac{3}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{5}{2}a^{12}+a^{11}-4a^{10}-\frac{1}{2}a^{9}+\frac{3}{2}a^{8}-\frac{5}{2}a^{7}+\frac{11}{2}a^{6}-a^{5}+\frac{3}{2}a^{4}+\frac{13}{2}a^{3}-\frac{9}{2}a^{2}+\frac{11}{2}a-3$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}+a^{11}+a^{10}-a^{8}-a^{7}-2a^{6}+\frac{3}{2}a^{4}+\frac{5}{2}a^{3}+\frac{3}{2}a^{2}-2a-1$, $2a^{22}+\frac{9}{2}a^{21}+4a^{20}+\frac{1}{2}a^{19}-4a^{18}-\frac{11}{2}a^{17}-\frac{9}{2}a^{16}-\frac{3}{2}a^{15}+\frac{3}{2}a^{14}+5a^{13}+8a^{12}+6a^{11}-\frac{1}{2}a^{10}-10a^{9}-\frac{25}{2}a^{8}-7a^{7}+\frac{5}{2}a^{6}+\frac{19}{2}a^{5}+\frac{21}{2}a^{4}+\frac{21}{2}a^{3}+5a^{2}-4a-5$, $a^{22}+\frac{1}{2}a^{21}+\frac{1}{2}a^{20}+2a^{19}-\frac{3}{2}a^{18}-\frac{3}{2}a^{17}+a^{16}+\frac{5}{2}a^{15}-2a^{14}-\frac{1}{2}a^{13}+\frac{1}{2}a^{12}+\frac{1}{2}a^{10}+\frac{5}{2}a^{9}-3a^{8}-\frac{9}{2}a^{7}+\frac{11}{2}a^{6}+4a^{5}-\frac{9}{2}a^{4}-5a^{3}+\frac{13}{2}a^{2}-\frac{5}{2}a+9$, $a^{22}+\frac{1}{2}a^{21}-a^{20}-a^{19}-\frac{1}{2}a^{18}+\frac{1}{2}a^{17}+\frac{3}{2}a^{16}+\frac{1}{2}a^{15}-a^{14}-\frac{3}{2}a^{13}-\frac{3}{2}a^{12}+2a^{11}+\frac{5}{2}a^{10}+a^{9}-\frac{9}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{2}a^{5}+\frac{5}{2}a^{4}+5a^{3}-\frac{9}{2}a^{2}-\frac{7}{2}a+3$, $a^{22}+a^{21}+a^{20}+\frac{3}{2}a^{19}+\frac{3}{2}a^{18}+a^{17}+a^{16}+\frac{1}{2}a^{15}-\frac{3}{2}a^{14}-3a^{13}-3a^{12}-3a^{11}-2a^{10}+a^{9}+\frac{5}{2}a^{8}+\frac{3}{2}a^{7}+2a^{6}+3a^{5}+\frac{3}{2}a^{4}+\frac{3}{2}a^{3}+3a^{2}+3$, $7a^{22}+\frac{7}{2}a^{21}+3a^{20}+a^{19}+a^{18}-a^{16}-a^{15}-2a^{14}-\frac{1}{2}a^{13}-a^{12}+\frac{1}{2}a^{10}+a^{9}+3a^{8}+2a^{7}+2a^{6}+a^{5}-a^{4}+2a^{3}-\frac{11}{2}a^{2}+a+39$, $13a^{22}+8a^{21}+\frac{5}{2}a^{20}+\frac{5}{2}a^{19}+\frac{3}{2}a^{18}-\frac{3}{2}a^{17}+\frac{3}{2}a^{16}+\frac{1}{2}a^{15}-\frac{5}{2}a^{14}+3a^{13}+a^{12}-3a^{11}+4a^{10}+\frac{1}{2}a^{9}-\frac{7}{2}a^{8}+\frac{9}{2}a^{7}-\frac{5}{2}a^{6}-\frac{9}{2}a^{5}+\frac{13}{2}a^{4}-\frac{11}{2}a^{3}-5a^{2}+11a+85$, $\frac{1}{2}a^{22}+2a^{21}+\frac{7}{2}a^{20}+\frac{3}{2}a^{19}-3a^{18}-\frac{13}{2}a^{17}-4a^{16}+2a^{14}-2a^{13}-\frac{7}{2}a^{12}-\frac{3}{2}a^{11}+6a^{10}+\frac{15}{2}a^{9}+\frac{13}{2}a^{8}+2a^{7}+\frac{5}{2}a^{6}-a^{5}-5a^{4}-10a^{3}-5a^{2}+\frac{3}{2}a+3$ (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: | \( 3129530905130 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 3129530905130 \cdot 1}{2\cdot\sqrt{2227978859786797789863805466564380466715471}}\cr\approx \mathstrut & 1.26328768877246 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 25852016738884976640000 |
The 1255 conjugacy class representatives for $S_{23}$ are not computed |
Character table for $S_{23}$ is not computed |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 46 sibling: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $20{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | $15{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.13.0.1}{13} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(661\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(102898348801\) | $\Q_{102898348801}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{102898348801}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(109\!\cdots\!137\) | $\Q_{10\!\cdots\!37}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ |