Normalized defining polynomial
\( x^{12} - 3 x^{11} + 129 x^{10} - 381 x^{9} + 7305 x^{8} - 20358 x^{7} + 228508 x^{6} - 535884 x^{5} + \cdots + 160801651 \)
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: |
\(215715056426865017578125\)
\(\medspace = 3^{16}\cdot 5^{9}\cdot 37^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(88.00\) | 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: | $3^{4/3}5^{3/4}37^{1/2}\approx 88.00149258596375$ | ||
Ramified primes: |
\(3\), \(5\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(1665=3^{2}\cdot 5\cdot 37\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1665}(1,·)$, $\chi_{1665}(1444,·)$, $\chi_{1665}(517,·)$, $\chi_{1665}(73,·)$, $\chi_{1665}(556,·)$, $\chi_{1665}(334,·)$, $\chi_{1665}(1072,·)$, $\chi_{1665}(628,·)$, $\chi_{1665}(1111,·)$, $\chi_{1665}(889,·)$, $\chi_{1665}(1627,·)$, $\chi_{1665}(1183,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | 4.0.171125.1$^{2}$, 12.0.215715056426865017578125.1$^{30}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{19}a^{8}+\frac{1}{19}a^{7}-\frac{5}{19}a^{6}-\frac{5}{19}a^{5}+\frac{9}{19}a^{4}-\frac{3}{19}a^{3}+\frac{1}{19}a^{2}+\frac{7}{19}a-\frac{6}{19}$, $\frac{1}{2071}a^{9}-\frac{30}{2071}a^{8}+\frac{154}{2071}a^{7}-\frac{876}{2071}a^{6}-\frac{64}{2071}a^{5}-\frac{909}{2071}a^{4}-\frac{514}{2071}a^{3}+\frac{1002}{2071}a^{2}+\frac{138}{2071}a-\frac{555}{2071}$, $\frac{1}{2071}a^{10}+\frac{17}{2071}a^{8}+\frac{365}{2071}a^{7}+\frac{906}{2071}a^{6}-\frac{431}{2071}a^{5}-\frac{207}{2071}a^{4}-\frac{139}{2071}a^{3}-\frac{104}{2071}a^{2}+\frac{642}{2071}a-\frac{518}{2071}$, $\frac{1}{72\!\cdots\!21}a^{11}-\frac{57\!\cdots\!70}{38\!\cdots\!59}a^{10}+\frac{94\!\cdots\!30}{72\!\cdots\!21}a^{9}-\frac{68\!\cdots\!19}{72\!\cdots\!21}a^{8}-\frac{49\!\cdots\!57}{72\!\cdots\!21}a^{7}+\frac{14\!\cdots\!12}{72\!\cdots\!21}a^{6}-\frac{36\!\cdots\!98}{72\!\cdots\!21}a^{5}+\frac{27\!\cdots\!98}{72\!\cdots\!21}a^{4}-\frac{23\!\cdots\!65}{72\!\cdots\!21}a^{3}-\frac{12\!\cdots\!77}{72\!\cdots\!21}a^{2}+\frac{16\!\cdots\!06}{38\!\cdots\!59}a+\frac{16\!\cdots\!77}{72\!\cdots\!21}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{14}\times C_{910}$, which has order $25480$ (assuming GRH)
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: |
$\frac{25484425413}{27\!\cdots\!69}a^{11}-\frac{86644940238}{27\!\cdots\!69}a^{10}+\frac{2629668375542}{27\!\cdots\!69}a^{9}-\frac{9138491661015}{27\!\cdots\!69}a^{8}+\frac{117154854295740}{27\!\cdots\!69}a^{7}-\frac{382042450140320}{27\!\cdots\!69}a^{6}+\frac{27\!\cdots\!93}{27\!\cdots\!69}a^{5}-\frac{62282902166052}{25\!\cdots\!41}a^{4}+\frac{32\!\cdots\!19}{27\!\cdots\!69}a^{3}-\frac{56\!\cdots\!83}{27\!\cdots\!69}a^{2}+\frac{86\!\cdots\!27}{14\!\cdots\!51}a-\frac{58\!\cdots\!25}{27\!\cdots\!69}$, $\frac{83\!\cdots\!55}{66\!\cdots\!69}a^{11}-\frac{46\!\cdots\!44}{66\!\cdots\!69}a^{10}+\frac{10\!\cdots\!10}{66\!\cdots\!69}a^{9}-\frac{62\!\cdots\!05}{66\!\cdots\!69}a^{8}+\frac{58\!\cdots\!10}{66\!\cdots\!69}a^{7}-\frac{32\!\cdots\!30}{66\!\cdots\!69}a^{6}+\frac{17\!\cdots\!33}{66\!\cdots\!69}a^{5}-\frac{86\!\cdots\!15}{66\!\cdots\!69}a^{4}+\frac{28\!\cdots\!65}{66\!\cdots\!69}a^{3}-\frac{10\!\cdots\!35}{66\!\cdots\!69}a^{2}+\frac{17\!\cdots\!90}{66\!\cdots\!69}a-\frac{30\!\cdots\!05}{35\!\cdots\!51}$, $\frac{19\!\cdots\!50}{66\!\cdots\!69}a^{11}+\frac{24\!\cdots\!24}{66\!\cdots\!69}a^{10}+\frac{69\!\cdots\!20}{66\!\cdots\!69}a^{9}+\frac{26\!\cdots\!65}{66\!\cdots\!69}a^{8}-\frac{53\!\cdots\!20}{66\!\cdots\!69}a^{7}+\frac{12\!\cdots\!50}{66\!\cdots\!69}a^{6}-\frac{41\!\cdots\!52}{66\!\cdots\!69}a^{5}+\frac{32\!\cdots\!65}{66\!\cdots\!69}a^{4}-\frac{10\!\cdots\!80}{66\!\cdots\!69}a^{3}+\frac{40\!\cdots\!75}{66\!\cdots\!69}a^{2}-\frac{76\!\cdots\!55}{66\!\cdots\!69}a+\frac{17\!\cdots\!71}{66\!\cdots\!69}$, $\frac{46\!\cdots\!67}{72\!\cdots\!21}a^{11}-\frac{49\!\cdots\!58}{72\!\cdots\!21}a^{10}+\frac{62\!\cdots\!98}{72\!\cdots\!21}a^{9}-\frac{52\!\cdots\!20}{72\!\cdots\!21}a^{8}+\frac{37\!\cdots\!40}{72\!\cdots\!21}a^{7}-\frac{23\!\cdots\!30}{72\!\cdots\!21}a^{6}+\frac{11\!\cdots\!05}{72\!\cdots\!21}a^{5}-\frac{48\!\cdots\!33}{66\!\cdots\!69}a^{4}+\frac{19\!\cdots\!91}{72\!\cdots\!21}a^{3}-\frac{59\!\cdots\!22}{72\!\cdots\!21}a^{2}+\frac{12\!\cdots\!12}{72\!\cdots\!21}a-\frac{34\!\cdots\!64}{72\!\cdots\!21}$, $\frac{80\!\cdots\!70}{72\!\cdots\!21}a^{11}-\frac{18\!\cdots\!47}{72\!\cdots\!21}a^{10}+\frac{89\!\cdots\!26}{72\!\cdots\!21}a^{9}-\frac{27\!\cdots\!20}{72\!\cdots\!21}a^{8}+\frac{40\!\cdots\!54}{72\!\cdots\!21}a^{7}-\frac{15\!\cdots\!17}{72\!\cdots\!21}a^{6}+\frac{89\!\cdots\!21}{72\!\cdots\!21}a^{5}-\frac{41\!\cdots\!45}{72\!\cdots\!21}a^{4}+\frac{98\!\cdots\!10}{72\!\cdots\!21}a^{3}-\frac{52\!\cdots\!89}{72\!\cdots\!21}a^{2}+\frac{46\!\cdots\!55}{72\!\cdots\!21}a-\frac{30\!\cdots\!17}{72\!\cdots\!21}$
| 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: | \( 201.000834787 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 201.000834787 \cdot 25480}{2\cdot\sqrt{215715056426865017578125}}\cr\approx \mathstrut & 0.339239387014 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 12 |
The 12 conjugacy class representatives for $C_{12}$ |
Character table for $C_{12}$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 4.0.171125.1, 6.6.820125.1 |
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 | ${\href{/padicField/2.12.0.1}{12} }$ | R | R | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.12.0.1}{12} }$ | ${\href{/padicField/17.4.0.1}{4} }^{3}$ | ${\href{/padicField/19.1.0.1}{1} }^{12}$ | ${\href{/padicField/23.12.0.1}{12} }$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.12.0.1}{12} }$ | ${\href{/padicField/47.12.0.1}{12} }$ | ${\href{/padicField/53.4.0.1}{4} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\)
| 3.12.16.14 | $x^{12} + 24 x^{11} + 216 x^{10} + 768 x^{9} - 432 x^{8} - 10368 x^{7} - 18414 x^{6} + 27864 x^{5} + 83592 x^{4} + 10800 x^{3} + 64800 x^{2} + 901125$ | $3$ | $4$ | $16$ | $C_{12}$ | $[2]^{4}$ |
\(5\)
| 5.12.9.1 | $x^{12} - 30 x^{8} + 225 x^{4} + 1125$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
\(37\)
| 37.4.2.2 | $x^{4} - 1221 x^{2} + 2738$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
37.4.2.2 | $x^{4} - 1221 x^{2} + 2738$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
37.4.2.2 | $x^{4} - 1221 x^{2} + 2738$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |