Normalized defining polynomial
\( x^{12} - x^{11} - 4 x^{10} - 10 x^{9} + 39 x^{8} - 78 x^{7} + 214 x^{6} - 280 x^{5} + 693 x^{4} - 573 x^{3} - 222 x^{2} + 123 x + 601 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(61132828589969773\) \(\medspace = 7^{8}\cdot 13^{9}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(25.05\) | 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: | $7^{2/3}13^{3/4}\approx 25.052796318948193$ | ||
Ramified primes: | \(7\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
$\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(91=7\cdot 13\) | ||
Dirichlet character group: | $\lbrace$$\chi_{91}(64,·)$, $\chi_{91}(1,·)$, $\chi_{91}(8,·)$, $\chi_{91}(44,·)$, $\chi_{91}(79,·)$, $\chi_{91}(18,·)$, $\chi_{91}(51,·)$, $\chi_{91}(53,·)$, $\chi_{91}(86,·)$, $\chi_{91}(25,·)$, $\chi_{91}(57,·)$, $\chi_{91}(60,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | 4.0.2197.1$^{2}$, 12.0.61132828589969773.1$^{30}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{57\!\cdots\!97}a^{11}-\frac{11\!\cdots\!61}{19\!\cdots\!99}a^{10}-\frac{16\!\cdots\!53}{19\!\cdots\!99}a^{9}-\frac{74\!\cdots\!94}{19\!\cdots\!99}a^{8}+\frac{63\!\cdots\!70}{19\!\cdots\!99}a^{7}-\frac{86\!\cdots\!97}{19\!\cdots\!99}a^{6}+\frac{10\!\cdots\!12}{57\!\cdots\!97}a^{5}-\frac{38\!\cdots\!46}{19\!\cdots\!99}a^{4}-\frac{51\!\cdots\!15}{57\!\cdots\!97}a^{3}-\frac{15\!\cdots\!32}{57\!\cdots\!97}a^{2}+\frac{46\!\cdots\!21}{19\!\cdots\!99}a-\frac{53\!\cdots\!00}{19\!\cdots\!99}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}$, which has order $4$
Unit group
Rank: | $5$ | 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: | $\frac{13505158842215}{19\!\cdots\!99}a^{11}-\frac{10034830535733}{19\!\cdots\!99}a^{10}-\frac{68577174796381}{19\!\cdots\!99}a^{9}-\frac{182849101111416}{19\!\cdots\!99}a^{8}+\frac{444597967255710}{19\!\cdots\!99}a^{7}-\frac{646265040804596}{19\!\cdots\!99}a^{6}+\frac{33\!\cdots\!45}{19\!\cdots\!99}a^{5}-\frac{20\!\cdots\!14}{19\!\cdots\!99}a^{4}+\frac{53\!\cdots\!56}{19\!\cdots\!99}a^{3}-\frac{55\!\cdots\!87}{19\!\cdots\!99}a^{2}-\frac{99\!\cdots\!57}{19\!\cdots\!99}a-\frac{17\!\cdots\!22}{19\!\cdots\!99}$, $\frac{18606448286018}{19\!\cdots\!99}a^{11}-\frac{22787130098526}{19\!\cdots\!99}a^{10}-\frac{77885140992422}{19\!\cdots\!99}a^{9}-\frac{125483082135645}{19\!\cdots\!99}a^{8}+\frac{815682105662352}{19\!\cdots\!99}a^{7}-\frac{16\!\cdots\!38}{19\!\cdots\!99}a^{6}+\frac{34\!\cdots\!80}{19\!\cdots\!99}a^{5}-\frac{49\!\cdots\!53}{19\!\cdots\!99}a^{4}+\frac{11\!\cdots\!38}{19\!\cdots\!99}a^{3}-\frac{58\!\cdots\!17}{19\!\cdots\!99}a^{2}-\frac{14\!\cdots\!37}{19\!\cdots\!99}a+\frac{92\!\cdots\!88}{19\!\cdots\!99}$, $\frac{4759225}{36677252641}a^{11}-\frac{48283509}{36677252641}a^{10}+\frac{15475111}{110031757923}a^{9}+\frac{260541560}{36677252641}a^{8}+\frac{621682972}{36677252641}a^{7}-\frac{2377721180}{36677252641}a^{6}+\frac{2115942755}{36677252641}a^{5}-\frac{5373201551}{36677252641}a^{4}+\frac{15538474330}{110031757923}a^{3}+\frac{1914167001}{36677252641}a^{2}-\frac{3276057308}{110031757923}a+\frac{18037614890}{110031757923}$, $\frac{19779208541530}{57\!\cdots\!97}a^{11}-\frac{45518170004191}{57\!\cdots\!97}a^{10}-\frac{62087869413088}{19\!\cdots\!99}a^{9}-\frac{91412709544312}{19\!\cdots\!99}a^{8}+\frac{459363543766811}{19\!\cdots\!99}a^{7}-\frac{167454570157973}{19\!\cdots\!99}a^{6}+\frac{45\!\cdots\!20}{57\!\cdots\!97}a^{5}-\frac{36\!\cdots\!23}{57\!\cdots\!97}a^{4}+\frac{20\!\cdots\!72}{57\!\cdots\!97}a^{3}-\frac{17\!\cdots\!26}{57\!\cdots\!97}a^{2}-\frac{35\!\cdots\!22}{57\!\cdots\!97}a+\frac{10\!\cdots\!28}{19\!\cdots\!99}$, $\frac{4385715924758}{19\!\cdots\!99}a^{11}-\frac{9944953612049}{57\!\cdots\!97}a^{10}-\frac{39176104594902}{19\!\cdots\!99}a^{9}-\frac{79292545692662}{19\!\cdots\!99}a^{8}+\frac{245119790197250}{19\!\cdots\!99}a^{7}-\frac{62553440982499}{19\!\cdots\!99}a^{6}+\frac{883354903803517}{19\!\cdots\!99}a^{5}-\frac{20\!\cdots\!98}{57\!\cdots\!97}a^{4}+\frac{40\!\cdots\!58}{19\!\cdots\!99}a^{3}-\frac{10\!\cdots\!64}{57\!\cdots\!97}a^{2}-\frac{21\!\cdots\!23}{57\!\cdots\!97}a-\frac{78\!\cdots\!49}{19\!\cdots\!99}$ | 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: | \( 562.775330001 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 562.775330001 \cdot 4}{2\cdot\sqrt{61132828589969773}}\cr\approx \mathstrut & 0.280096067652 \end{aligned}\]
Galois group
A cyclic group of order 12 |
The 12 conjugacy class representatives for $C_{12}$ |
Character table for $C_{12}$ |
Intermediate fields
\(\Q(\sqrt{13}) \), \(\Q(\zeta_{7})^+\), 4.0.2197.1, 6.6.5274997.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.12.0.1}{12} }$ | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | ${\href{/padicField/5.12.0.1}{12} }$ | R | ${\href{/padicField/11.12.0.1}{12} }$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.1.0.1}{1} }^{12}$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.12.0.1}{12} }$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.12.0.1}{12} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.12.8.1 | $x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
\(13\) | 13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.13.4t1.a.a | $1$ | $ 13 $ | 4.0.2197.1 | $C_4$ (as 4T1) | $0$ | $-1$ |
* | 1.13.2t1.a.a | $1$ | $ 13 $ | \(\Q(\sqrt{13}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.13.4t1.a.b | $1$ | $ 13 $ | 4.0.2197.1 | $C_4$ (as 4T1) | $0$ | $-1$ |
* | 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.91.12t1.a.a | $1$ | $ 7 \cdot 13 $ | 12.0.61132828589969773.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ |
* | 1.91.6t1.j.a | $1$ | $ 7 \cdot 13 $ | 6.6.5274997.1 | $C_6$ (as 6T1) | $0$ | $1$ |
* | 1.91.12t1.a.b | $1$ | $ 7 \cdot 13 $ | 12.0.61132828589969773.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ |
* | 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.91.12t1.a.c | $1$ | $ 7 \cdot 13 $ | 12.0.61132828589969773.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ |
* | 1.91.6t1.j.b | $1$ | $ 7 \cdot 13 $ | 6.6.5274997.1 | $C_6$ (as 6T1) | $0$ | $1$ |
* | 1.91.12t1.a.d | $1$ | $ 7 \cdot 13 $ | 12.0.61132828589969773.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ |