Normalized defining polynomial
\( x^{12} + 27 x^{10} - 20 x^{9} + 324 x^{8} - 210 x^{7} + 1968 x^{6} - 630 x^{5} + 5661 x^{4} - 520 x^{3} + 7380 x^{2} + 3280 \)
Invariants
| Degree: | $12$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(446676160500000000=2^{8}\cdot 3^{12}\cdot 5^{9}\cdot 41^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $29.57$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 3, 5, 41$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{3}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{43899652739570864} a^{11} - \frac{999690133719313}{21949826369785432} a^{10} - \frac{2585030706369389}{43899652739570864} a^{9} + \frac{6913551788429743}{21949826369785432} a^{8} + \frac{5223756891550717}{10974913184892716} a^{7} - \frac{1032874728445673}{21949826369785432} a^{6} - \frac{5458510751125139}{10974913184892716} a^{5} - \frac{3894956792769587}{21949826369785432} a^{4} - \frac{6436445058509031}{43899652739570864} a^{3} - \frac{3363860063046473}{21949826369785432} a^{2} + \frac{972370001877287}{10974913184892716} a + \frac{951882889208629}{2743728296223179}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $5$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{15500786844339}{21949826369785432} a^{11} + \frac{27751140888123}{21949826369785432} a^{10} + \frac{375609142391647}{21949826369785432} a^{9} + \frac{406838397290229}{21949826369785432} a^{8} + \frac{1713425502939873}{10974913184892716} a^{7} + \frac{2848304050558959}{10974913184892716} a^{6} + \frac{7185128607484389}{10974913184892716} a^{5} + \frac{20488579510413213}{10974913184892716} a^{4} + \frac{25054598797503653}{21949826369785432} a^{3} + \frac{98360570878469451}{21949826369785432} a^{2} + \frac{15072739802154837}{10974913184892716} a + \frac{19199617478949547}{5487456592446358} \) (order $10$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 27699.7184504 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 648 |
| The 30 conjugacy class representatives for [3^4:2]4 |
| Character table for [3^4:2]4 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | ${\href{/LocalNumberField/7.12.0.1}{12} }$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/13.12.0.1}{12} }$ | ${\href{/LocalNumberField/17.12.0.1}{12} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/23.12.0.1}{12} }$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/37.12.0.1}{12} }$ | R | ${\href{/LocalNumberField/43.12.0.1}{12} }$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.0.1 | $x^{4} - x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 2.8.8.1 | $x^{8} + 28 x^{4} + 144$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
| $3$ | 3.12.12.7 | $x^{12} + 12 x^{11} + 108 x^{10} + 105 x^{9} - 45 x^{8} + 45 x^{7} - 81 x^{6} - 108 x^{5} + 27 x^{4} - 81 x^{3} - 81 x^{2} + 81 x - 81$ | $3$ | $4$ | $12$ | 12T119 | $[3/2, 3/2, 3/2]_{2}^{4}$ |
| $5$ | 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $41$ | $\Q_{41}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{41}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{41}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.3.2.1 | $x^{3} - 41$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |