Normalized defining polynomial
\( x^{23} - 7 x^{22} + 2 x^{21} + 88 x^{20} - 150 x^{19} - 422 x^{18} + 1148 x^{17} + 862 x^{16} - 4167 x^{15} - 105 x^{14} + 8563 x^{13} - 2935 x^{12} - 10393 x^{11} + 5920 x^{10} + 7285 x^{9} + \cdots - 1 \)
Invariants
Degree: | $23$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[13, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-425103981715298419969372518134528837127343\) \(\medspace = -\,13\cdot 1289\cdot 4057217\cdot 26391289\cdot 32505503\cdot 7288751994309941\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(64.56\) | 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: | $13^{1/2}1289^{1/2}4057217^{1/2}26391289^{1/2}32505503^{1/2}7288751994309941^{1/2}\approx 6.519999859779894e+20$ | ||
Ramified primes: | \(13\), \(1289\), \(4057217\), \(26391289\), \(32505503\), \(7288751994309941\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-42510\!\cdots\!27343}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $17$ | 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: | $a$, $a-1$, $56a^{22}-367a^{21}-52a^{20}+4906a^{19}-6211a^{18}-26417a^{17}+52525a^{16}+71758a^{15}-201497a^{14}-95821a^{13}+437252a^{12}+30524a^{11}-569112a^{10}+78197a^{9}+443407a^{8}-105522a^{7}-196276a^{6}+52962a^{5}+43561a^{4}-10725a^{3}-3710a^{2}+536a+126$, $a^{2}-1$, $a^{22}-7a^{21}+2a^{20}+88a^{19}-150a^{18}-422a^{17}+1148a^{16}+862a^{15}-4167a^{14}-105a^{13}+8563a^{12}-2935a^{11}-10393a^{10}+5920a^{9}+7285a^{8}-5405a^{7}-2661a^{6}+2503a^{5}+355a^{4}-536a^{3}+23a^{2}+36a-4$, $25a^{22}-164a^{21}-22a^{20}+2189a^{19}-2785a^{18}-11763a^{17}+23486a^{16}+31855a^{15}-89941a^{14}-42276a^{13}+194884a^{12}+12896a^{11}-253323a^{10}+35447a^{9}+197158a^{8}-47260a^{7}-87205a^{6}+23623a^{5}+19343a^{4}-4767a^{3}-1645a^{2}+235a+56$, $5a^{21}-34a^{20}+4a^{19}+435a^{18}-660a^{17}-2173a^{16}+5173a^{15}+5043a^{14}-18893a^{13}-3813a^{12}+38918a^{11}-6529a^{10}-47409a^{9}+17532a^{8}+33627a^{7}-16302a^{6}-12780a^{5}+7072a^{4}+2046a^{3}-1270a^{2}-27a+43$, $4a^{22}-24a^{21}-17a^{20}+341a^{19}-254a^{18}-2033a^{17}+2629a^{16}+6639a^{15}-10831a^{14}-13046a^{13}+24710a^{12}+16030a^{11}-33742a^{10}-12443a^{9}+27692a^{8}+5967a^{7}-13008a^{6}-1564a^{5}+3131a^{4}+144a^{3}-313a^{2}-2a+9$, $a^{22}-9a^{21}+16a^{20}+84a^{19}-326a^{18}-122a^{17}+1992a^{16}-1434a^{15}-5890a^{14}+8224a^{13}+8770a^{12}-20013a^{11}-4553a^{10}+26530a^{9}-4368a^{8}-19664a^{7}+7736a^{6}+7548a^{5}-4217a^{4}-1132a^{3}+874a^{2}-13a-34$, $15a^{22}-101a^{21}+4a^{20}+1315a^{19}-1900a^{18}-6757a^{17}+15312a^{16}+16601a^{15}-57193a^{14}-15814a^{13}+120883a^{12}-12707a^{11}-152192a^{10}+47211a^{9}+112983a^{8}-47593a^{7}-46132a^{6}+21847a^{5}+8646a^{4}-4168a^{3}-425a^{2}+170a+15$, $5a^{22}-38a^{21}+31a^{20}+434a^{19}-1014a^{18}-1660a^{17}+7006a^{16}+866a^{15}-23421a^{14}+11976a^{13}+43129a^{12}-40360a^{11}-43154a^{10}+60740a^{9}+18665a^{8}-48736a^{7}+2842a^{6}+20249a^{5}-5559a^{4}-3554a^{3}+1547a^{2}+81a-72$, $79a^{22}-519a^{21}-66a^{20}+6928a^{19}-8870a^{18}-37207a^{17}+74769a^{16}+100501a^{15}-286562a^{14}-132015a^{13}+621671a^{12}+35634a^{11}-809286a^{10}+120599a^{9}+631048a^{8}-157233a^{7}-279955a^{6}+78450a^{5}+62502a^{4}-15902a^{3}-5420a^{2}+801a+187$, $28a^{22}-190a^{21}+17a^{20}+2455a^{19}-3666a^{18}-12453a^{17}+29169a^{16}+29755a^{15}-108140a^{14}-25215a^{13}+226974a^{12}-32055a^{11}-283529a^{10}+97523a^{9}+208412a^{8}-94902a^{7}-83865a^{6}+42849a^{5}+15254a^{4}-8090a^{3}-653a^{2}+327a+24$, $7a^{22}-45a^{21}-11a^{20}+606a^{19}-710a^{18}-3301a^{17}+6145a^{16}+9154a^{15}-23671a^{14}-12837a^{13}+51180a^{12}+5650a^{11}-65791a^{10}+7690a^{9}+49871a^{8}-11978a^{7}-20848a^{6}+6348a^{5}+4096a^{4}-1342a^{3}-263a^{2}+61a+10$, $45a^{22}-301a^{21}-4a^{20}+3961a^{19}-5517a^{18}-20736a^{17}+45240a^{16}+53025a^{15}-171283a^{14}-58468a^{13}+368124a^{12}-16844a^{11}-474914a^{10}+118123a^{9}+366871a^{8}-129029a^{7}-161306a^{6}+62589a^{5}+35872a^{4}-13048a^{3}-3224a^{2}+743a+142$, $10a^{22}-65a^{21}-10a^{20}+859a^{19}-1074a^{18}-4547a^{17}+8963a^{16}+11982a^{15}-33644a^{14}-14806a^{13}+70640a^{12}+1604a^{11}-87294a^{10}+17941a^{9}+62317a^{8}-21446a^{7}-23380a^{6}+10514a^{5}+3483a^{4}-2089a^{3}+30a^{2}+80a-7$, $11a^{22}-75a^{21}+9a^{20}+966a^{19}-1479a^{18}-4851a^{17}+11715a^{16}+11210a^{15}-43311a^{14}-7631a^{13}+90396a^{12}-18460a^{11}-111483a^{10}+47088a^{9}+79648a^{8}-44843a^{7}-29984a^{6}+20334a^{5}+4442a^{4}-3887a^{3}+50a^{2}+152a+1$ (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: | \( 4090939159600 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{13}\cdot(2\pi)^{5}\cdot 4090939159600 \cdot 1}{2\cdot\sqrt{425103981715298419969372518134528837127343}}\cr\approx \mathstrut & 0.251671897827095 \end{aligned}\] (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 | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | $18{,}\,{\href{/padicField/3.5.0.1}{5} }$ | $22{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.13.0.1}{13} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $19{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/31.3.0.1}{3} }$ | $20{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.9.0.1}{9} }$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
13.8.0.1 | $x^{8} + 8 x^{4} + 12 x^{3} + 2 x^{2} + 3 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
13.13.0.1 | $x^{13} + 12 x + 11$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
\(1289\) | $\Q_{1289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(4057217\) | $\Q_{4057217}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
\(26391289\) | $\Q_{26391289}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
\(32505503\) | 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 $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
\(7288751994309941\) | $\Q_{7288751994309941}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{7288751994309941}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{7288751994309941}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{7288751994309941}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
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 $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |