Normalized defining polynomial
\( x^{23} - 3 x^{21} - 6 x^{20} + x^{19} + 17 x^{18} + 18 x^{17} - 10 x^{16} - 43 x^{15} - 25 x^{14} + 33 x^{13} + 60 x^{12} + 11 x^{11} - 54 x^{10} - 47 x^{9} + 12 x^{8} + 43 x^{7} + 17 x^{6} - 17 x^{5} + \cdots - 1 \)
Invariants
Degree: | $23$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[5, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-1377082951373957855841789612501108908\) \(\medspace = -\,2^{2}\cdot 641\cdot 31357146703\cdot 17127956171402395614949\) | 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: | \(37.26\) | 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: | $2^{2/3}641^{1/2}31357146703^{1/2}17127956171402395614949^{1/2}\approx 9.314010408753599e+17$ | ||
Ramified primes: | \(2\), \(641\), \(31357146703\), \(17127956171402395614949\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-34427\!\cdots\!77227}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $13$ | 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: | $a^{22}-3a^{20}-6a^{19}+a^{18}+17a^{17}+18a^{16}-10a^{15}-43a^{14}-25a^{13}+33a^{12}+60a^{11}+11a^{10}-54a^{9}-47a^{8}+12a^{7}+43a^{6}+17a^{5}-17a^{4}-16a^{3}+6a+1$, $a^{22}-3a^{20}-6a^{19}+a^{18}+17a^{17}+18a^{16}-10a^{15}-43a^{14}-25a^{13}+33a^{12}+60a^{11}+11a^{10}-54a^{9}-47a^{8}+12a^{7}+43a^{6}+17a^{5}-16a^{4}-16a^{3}-a^{2}+5a$, $a^{22}-3a^{20}-6a^{19}+a^{18}+17a^{17}+18a^{16}-10a^{15}-43a^{14}-25a^{13}+33a^{12}+60a^{11}+11a^{10}-54a^{9}-47a^{8}+12a^{7}+43a^{6}+18a^{5}-17a^{4}-17a^{3}-2a^{2}+5a+2$, $5a^{22}+a^{21}-14a^{20}-33a^{19}-4a^{18}+80a^{17}+108a^{16}-15a^{15}-207a^{14}-178a^{13}+97a^{12}+308a^{11}+148a^{10}-199a^{9}-278a^{8}-40a^{7}+180a^{6}+138a^{5}-27a^{4}-79a^{3}-30a^{2}+15a+9$, $4a^{22}+a^{21}-11a^{20}-27a^{19}-5a^{18}+63a^{17}+90a^{16}-5a^{15}-164a^{14}-153a^{13}+64a^{12}+248a^{11}+137a^{10}-145a^{9}-231a^{8}-53a^{7}+137a^{6}+122a^{5}-8a^{4}-63a^{3}-30a^{2}+8a+8$, $a^{22}+a^{21}-2a^{20}-9a^{19}-8a^{18}+13a^{17}+35a^{16}+22a^{15}-38a^{14}-71a^{13}-21a^{12}+70a^{11}+81a^{10}-9a^{9}-78a^{8}-52a^{7}+25a^{6}+45a^{5}+13a^{4}-15a^{3}-11a^{2}+1$, $7a^{22}+4a^{21}-19a^{20}-53a^{19}-22a^{18}+108a^{17}+187a^{16}+31a^{15}-287a^{14}-333a^{13}+53a^{12}+451a^{11}+320a^{10}-206a^{9}-440a^{8}-151a^{7}+216a^{6}+234a^{5}+8a^{4}-104a^{3}-57a^{2}+10a+12$, $3a^{22}-2a^{21}-7a^{20}-12a^{19}+10a^{18}+37a^{17}+20a^{16}-38a^{15}-75a^{14}+2a^{13}+82a^{12}+70a^{11}-46a^{10}-97a^{9}-16a^{8}+58a^{7}+46a^{6}-18a^{5}-35a^{4}-4a^{3}+9a^{2}+8a-4$, $4a^{22}+a^{21}-10a^{20}-25a^{19}-7a^{18}+51a^{17}+77a^{16}+8a^{15}-115a^{14}-116a^{13}+23a^{12}+149a^{11}+94a^{10}-67a^{9}-122a^{8}-43a^{7}+49a^{6}+54a^{5}+5a^{4}-18a^{3}-11a^{2}+4a+2$, $4a^{22}+4a^{21}-12a^{20}-35a^{19}-20a^{18}+69a^{17}+134a^{16}+33a^{15}-195a^{14}-254a^{13}+22a^{12}+329a^{11}+259a^{10}-139a^{9}-344a^{8}-129a^{7}+167a^{6}+193a^{5}+12a^{4}-91a^{3}-48a^{2}+9a+13$, $8a^{22}+6a^{21}-21a^{20}-65a^{19}-36a^{18}+122a^{17}+240a^{16}+71a^{15}-337a^{14}-452a^{13}+7a^{12}+563a^{11}+471a^{10}-206a^{9}-587a^{8}-257a^{7}+259a^{6}+331a^{5}+37a^{4}-138a^{3}-83a^{2}+9a+17$, $4a^{22}+2a^{21}-11a^{20}-28a^{19}-11a^{18}+58a^{17}+96a^{16}+15a^{15}-144a^{14}-165a^{13}+23a^{12}+213a^{11}+159a^{10}-87a^{9}-204a^{8}-86a^{7}+84a^{6}+113a^{5}+19a^{4}-44a^{3}-33a^{2}+a+7$, $7a^{22}-5a^{21}-16a^{20}-29a^{19}+26a^{18}+90a^{17}+48a^{16}-100a^{15}-195a^{14}+8a^{13}+225a^{12}+199a^{11}-134a^{10}-282a^{9}-53a^{8}+186a^{7}+155a^{6}-60a^{5}-120a^{4}-14a^{3}+41a^{2}+25a-15$ (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: | \( 3910844212.61 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{9}\cdot 3910844212.61 \cdot 1}{2\cdot\sqrt{1377082951373957855841789612501108908}}\cr\approx \mathstrut & 0.813821695340 \end{aligned}\] (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 | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | $19{,}\,{\href{/padicField/5.4.0.1}{4} }$ | $16{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | $19{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | $17{,}\,{\href{/padicField/31.6.0.1}{6} }$ | $17{,}\,{\href{/padicField/37.6.0.1}{6} }$ | $23$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $23$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.9.0.1}{9} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
2.12.0.1 | $x^{12} + x^{7} + x^{6} + x^{5} + x^{3} + x + 1$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(641\) | $\Q_{641}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(31357146703\) | $\Q_{31357146703}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(171\!\cdots\!949\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ |