Normalized defining polynomial
\( x^{16} - 5 x^{15} + 31 x^{13} - 8 x^{12} - 101 x^{11} + 7 x^{10} + 183 x^{9} + 32 x^{8} - 183 x^{7} - 61 x^{6} + 112 x^{5} + 78 x^{4} - 9 x^{3} - 28 x^{2} - 7 x + 7 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(30853268336830129281\) \(\medspace = 3^{8}\cdot 7^{8}\cdot 13^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(16.52\) | 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: | $3^{1/2}7^{2/3}13^{2/3}\approx 35.04194650073235$ | ||
Ramified primes: | \(3\), \(7\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7}a^{13}+\frac{1}{7}a^{12}+\frac{2}{7}a^{11}+\frac{3}{7}a^{10}+\frac{1}{7}a^{9}+\frac{3}{7}a^{8}-\frac{3}{7}a^{7}-\frac{2}{7}a^{6}+\frac{1}{7}a^{5}+\frac{2}{7}a^{4}-\frac{2}{7}a^{3}$, $\frac{1}{14}a^{14}+\frac{1}{14}a^{12}-\frac{3}{7}a^{11}+\frac{5}{14}a^{10}+\frac{1}{7}a^{9}-\frac{3}{7}a^{8}+\frac{1}{14}a^{7}+\frac{3}{14}a^{6}+\frac{1}{14}a^{5}+\frac{3}{14}a^{4}+\frac{1}{7}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{201188162}a^{15}-\frac{2928463}{100594081}a^{14}+\frac{8434785}{201188162}a^{13}+\frac{28114571}{100594081}a^{12}-\frac{98515681}{201188162}a^{11}+\frac{25976444}{100594081}a^{10}-\frac{7398143}{100594081}a^{9}-\frac{59162769}{201188162}a^{8}-\frac{810261}{1991962}a^{7}-\frac{38192121}{201188162}a^{6}-\frac{20136635}{201188162}a^{5}-\frac{9431326}{100594081}a^{4}+\frac{87385559}{201188162}a^{3}+\frac{2754567}{14370583}a^{2}+\frac{9175725}{28741166}a+\frac{177689}{14370583}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $7$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{2412144}{14370583} a^{15} - \frac{14391461}{14370583} a^{14} + \frac{15876474}{14370583} a^{13} + \frac{49247605}{14370583} a^{12} - \frac{65201610}{14370583} a^{11} - \frac{117821061}{14370583} a^{10} + \frac{93665114}{14370583} a^{9} + \frac{169284546}{14370583} a^{8} - \frac{48574}{142283} a^{7} - \frac{146349640}{14370583} a^{6} - \frac{65202436}{14370583} a^{5} + \frac{89113643}{14370583} a^{4} + \frac{104995800}{14370583} a^{3} + \frac{10704428}{14370583} a^{2} + \frac{3108868}{14370583} a - \frac{5893873}{14370583} \) (order $6$) | 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: | $\frac{46989633}{100594081}a^{15}-\frac{39451927}{14370583}a^{14}+\frac{233828640}{100594081}a^{13}+\frac{1311067369}{100594081}a^{12}-\frac{234728392}{14370583}a^{11}-\frac{3385791622}{100594081}a^{10}+\frac{3730221253}{100594081}a^{9}+\frac{5247556999}{100594081}a^{8}-\frac{5336050}{142283}a^{7}-\frac{4806752718}{100594081}a^{6}+\frac{1681630089}{100594081}a^{5}+\frac{3114844872}{100594081}a^{4}+\frac{944429679}{100594081}a^{3}-\frac{115289247}{14370583}a^{2}-\frac{86059412}{14370583}a+\frac{18862201}{14370583}$, $\frac{24630651}{100594081}a^{15}-\frac{269122433}{201188162}a^{14}+\frac{49257671}{100594081}a^{13}+\frac{1664320717}{201188162}a^{12}-\frac{744423399}{100594081}a^{11}-\frac{4720848071}{201188162}a^{10}+\frac{2007668504}{100594081}a^{9}+\frac{3819051272}{100594081}a^{8}-\frac{49785665}{1991962}a^{7}-\frac{6781573229}{201188162}a^{6}+\frac{3286029769}{201188162}a^{5}+\frac{576310713}{28741166}a^{4}+\frac{69506036}{100594081}a^{3}-\frac{124722977}{28741166}a^{2}-\frac{38071239}{14370583}a+\frac{66918641}{28741166}$, $\frac{8251872}{100594081}a^{15}-\frac{81856967}{201188162}a^{14}-\frac{2946340}{100594081}a^{13}+\frac{549009973}{201188162}a^{12}-\frac{140187044}{100594081}a^{11}-\frac{1443641963}{201188162}a^{10}+\frac{171683581}{100594081}a^{9}+\frac{1148978596}{100594081}a^{8}+\frac{1148229}{1991962}a^{7}-\frac{1670939181}{201188162}a^{6}-\frac{580748745}{201188162}a^{5}+\frac{860579271}{201188162}a^{4}+\frac{364009693}{100594081}a^{3}+\frac{52503247}{28741166}a^{2}-\frac{12261589}{14370583}a-\frac{12875141}{28741166}$, $\frac{25369882}{100594081}a^{15}-\frac{286934267}{201188162}a^{14}+\frac{11470186}{14370583}a^{13}+\frac{1641051961}{201188162}a^{12}-\frac{858188525}{100594081}a^{11}-\frac{4541461495}{201188162}a^{10}+\frac{2217399390}{100594081}a^{9}+\frac{527842621}{14370583}a^{8}-\frac{52963179}{1991962}a^{7}-\frac{6854141879}{201188162}a^{6}+\frac{3343246171}{201188162}a^{5}+\frac{4275719891}{201188162}a^{4}+\frac{70681136}{100594081}a^{3}-\frac{210930805}{28741166}a^{2}-\frac{59103262}{14370583}a+\frac{62531631}{28741166}$, $\frac{15306591}{100594081}a^{15}-\frac{11814331}{14370583}a^{14}+\frac{14755659}{100594081}a^{13}+\frac{592165758}{100594081}a^{12}-\frac{550357996}{100594081}a^{11}-\frac{241567021}{14370583}a^{10}+\frac{1716211177}{100594081}a^{9}+\frac{2716985790}{100594081}a^{8}-\frac{24887279}{995981}a^{7}-\frac{342534968}{14370583}a^{6}+\frac{283710550}{14370583}a^{5}+\frac{1326600139}{100594081}a^{4}-\frac{528375770}{100594081}a^{3}-\frac{31541984}{14370583}a^{2}-\frac{12553567}{14370583}a+\frac{15136906}{14370583}$, $\frac{34969023}{201188162}a^{15}-\frac{267070075}{201188162}a^{14}+\frac{606895707}{201188162}a^{13}+\frac{162331049}{201188162}a^{12}-\frac{1977265495}{201188162}a^{11}+\frac{575609287}{201188162}a^{10}+\frac{235879479}{14370583}a^{9}-\frac{1610063225}{201188162}a^{8}-\frac{12079496}{995981}a^{7}+\frac{587031092}{100594081}a^{6}+\frac{409350481}{100594081}a^{5}-\frac{289581377}{201188162}a^{4}+\frac{210357975}{201188162}a^{3}-\frac{27431237}{28741166}a^{2}+\frac{48754335}{28741166}a-\frac{41975799}{28741166}$, $\frac{25843939}{201188162}a^{15}-\frac{127079777}{201188162}a^{14}+\frac{814733}{28741166}a^{13}+\frac{723526581}{201188162}a^{12}-\frac{23463341}{28741166}a^{11}-\frac{2096540653}{201188162}a^{10}-\frac{56012529}{100594081}a^{9}+\frac{3101679557}{201188162}a^{8}+\frac{5856348}{995981}a^{7}-\frac{985955228}{100594081}a^{6}-\frac{637940836}{100594081}a^{5}+\frac{716074859}{201188162}a^{4}+\frac{1192809365}{201188162}a^{3}+\frac{94172601}{28741166}a^{2}+\frac{69500623}{28741166}a+\frac{55905727}{28741166}$ | 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: | \( 7145.924896370814 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 7145.924896370814 \cdot 1}{6\cdot\sqrt{30853268336830129281}}\cr\approx \mathstrut & 0.520829477092895 \end{aligned}\]
Galois group
$\SL(2,3):C_2$ (as 16T60):
A solvable group of order 48 |
The 14 conjugacy class representatives for $\SL(2,3):C_2$ |
Character table for $\SL(2,3):C_2$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 4.0.74529.1, 8.0.5554571841.6 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 24 sibling: | deg 24 |
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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.4.0.1}{4} }$ | R | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.4.0.1}{4} }$ | R | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.6.4.3 | $x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
13.6.4.3 | $x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.91.3t1.a.a | $1$ | $ 7 \cdot 13 $ | 3.3.8281.2 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.273.6t1.d.a | $1$ | $ 3 \cdot 7 \cdot 13 $ | 6.0.1851523947.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
1.91.3t1.a.b | $1$ | $ 7 \cdot 13 $ | 3.3.8281.2 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.273.6t1.d.b | $1$ | $ 3 \cdot 7 \cdot 13 $ | 6.0.1851523947.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
2.24843.24t21.a.a | $2$ | $ 3 \cdot 7^{2} \cdot 13^{2}$ | 16.0.30853268336830129281.2 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ | |
2.24843.24t21.a.b | $2$ | $ 3 \cdot 7^{2} \cdot 13^{2}$ | 16.0.30853268336830129281.2 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ | |
* | 2.273.16t60.a.a | $2$ | $ 3 \cdot 7 \cdot 13 $ | 16.0.30853268336830129281.2 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ |
* | 2.273.16t60.a.b | $2$ | $ 3 \cdot 7 \cdot 13 $ | 16.0.30853268336830129281.2 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ |
* | 2.273.16t60.a.c | $2$ | $ 3 \cdot 7 \cdot 13 $ | 16.0.30853268336830129281.2 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ |
* | 2.273.16t60.a.d | $2$ | $ 3 \cdot 7 \cdot 13 $ | 16.0.30853268336830129281.2 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ |
* | 3.24843.6t6.a.a | $3$ | $ 3 \cdot 7^{2} \cdot 13^{2}$ | 6.4.205724883.1 | $A_4\times C_2$ (as 6T6) | $1$ | $1$ |
* | 3.74529.4t4.a.a | $3$ | $ 3^{2} \cdot 7^{2} \cdot 13^{2}$ | 4.0.74529.1 | $A_4$ (as 4T4) | $1$ | $-1$ |