Normalized defining polynomial
\( x^{12} - 35 x^{10} - 52 x^{9} + 308 x^{8} + 808 x^{7} - 249 x^{6} - 2810 x^{5} - 3596 x^{4} - 1830 x^{3} + \cdots + 9 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(734661949275345164544\) \(\medspace = 2^{8}\cdot 3^{6}\cdot 89^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(54.81\) | 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))
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Galois root discriminant: | $2^{2/3}3^{1/2}89^{4/5}\approx 99.71472513477265$ | ||
Ramified primes: | \(2\), \(3\), \(89\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{6}a^{8}-\frac{1}{6}a^{7}+\frac{1}{6}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{6}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{18}a^{9}-\frac{1}{18}a^{8}+\frac{2}{9}a^{7}-\frac{1}{18}a^{6}+\frac{1}{9}a^{5}+\frac{1}{6}a^{4}-\frac{5}{18}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{36}a^{10}-\frac{1}{36}a^{9}+\frac{1}{36}a^{8}+\frac{1}{18}a^{7}-\frac{1}{36}a^{6}+\frac{5}{12}a^{5}-\frac{11}{36}a^{4}+\frac{1}{3}a^{3}-\frac{5}{12}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{180}a^{11}-\frac{1}{90}a^{10}+\frac{1}{45}a^{9}+\frac{1}{36}a^{8}-\frac{37}{180}a^{7}+\frac{1}{90}a^{6}-\frac{7}{45}a^{5}+\frac{41}{180}a^{4}+\frac{17}{180}a^{3}-\frac{3}{10}a^{2}+\frac{11}{30}a-\frac{7}{20}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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)
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Fundamental units: | $a+1$, $\frac{115}{18}a^{11}-\frac{185}{36}a^{10}-\frac{2633}{12}a^{9}-\frac{1871}{12}a^{8}+\frac{37645}{18}a^{7}+\frac{125375}{36}a^{6}-\frac{157487}{36}a^{5}-\frac{57759}{4}a^{4}-\frac{205543}{18}a^{3}-\frac{30437}{12}a^{2}+\frac{4415}{12}a+\frac{301}{4}$, $\frac{115}{18}a^{11}-\frac{185}{36}a^{10}-\frac{2633}{12}a^{9}-\frac{1871}{12}a^{8}+\frac{37645}{18}a^{7}+\frac{125375}{36}a^{6}-\frac{157487}{36}a^{5}-\frac{57759}{4}a^{4}-\frac{205543}{18}a^{3}-\frac{30437}{12}a^{2}+\frac{4415}{12}a+\frac{305}{4}$, $\frac{109}{18}a^{11}-\frac{43}{9}a^{10}-\frac{1249}{6}a^{9}-\frac{452}{3}a^{8}+\frac{35707}{18}a^{7}+3328a^{6}-\frac{24785}{6}a^{5}-\frac{123815}{9}a^{4}-\frac{196781}{18}a^{3}-2461a^{2}+\frac{2021}{6}a+70$, $\frac{133}{30}a^{11}-\frac{16}{5}a^{10}-\frac{6892}{45}a^{9}-\frac{2147}{18}a^{8}+\frac{131467}{90}a^{7}+\frac{226853}{90}a^{6}-\frac{270967}{90}a^{5}-\frac{103089}{10}a^{4}-\frac{371206}{45}a^{3}-\frac{9407}{5}a^{2}+\frac{7193}{30}a+\frac{266}{5}$, $\frac{713}{180}a^{11}-\frac{481}{180}a^{10}-\frac{24673}{180}a^{9}-\frac{2033}{18}a^{8}+\frac{234619}{180}a^{7}+\frac{416821}{180}a^{6}-\frac{471119}{180}a^{5}-\frac{422378}{45}a^{4}-\frac{1388309}{180}a^{3}-\frac{109129}{60}a^{2}+\frac{13421}{60}a+\frac{497}{10}$, $\frac{377}{30}a^{11}-\frac{1619}{180}a^{10}-\frac{77987}{180}a^{9}-\frac{12371}{36}a^{8}+\frac{370103}{90}a^{7}+\frac{432743}{60}a^{6}-\frac{1485191}{180}a^{5}-\frac{5288753}{180}a^{4}-\frac{2184053}{90}a^{3}-\frac{115857}{20}a^{2}+\frac{42619}{60}a+\frac{3191}{20}$, $\frac{154}{45}a^{11}-\frac{637}{180}a^{10}-\frac{2309}{20}a^{9}-\frac{721}{12}a^{8}+\frac{49297}{45}a^{7}+\frac{296287}{180}a^{6}-\frac{424273}{180}a^{5}-\frac{425983}{60}a^{4}-\frac{248512}{45}a^{3}-\frac{75253}{60}a^{2}+\frac{9817}{60}a+\frac{703}{20}$, $\frac{23}{30}a^{11}+\frac{59}{180}a^{10}-\frac{5033}{180}a^{9}-\frac{1775}{36}a^{8}+\frac{11491}{45}a^{7}+\frac{128281}{180}a^{6}-\frac{16423}{60}a^{5}-\frac{452227}{180}a^{4}-\frac{27623}{10}a^{3}-\frac{59759}{60}a^{2}+\frac{137}{20}a+\frac{799}{20}$, $\frac{13}{15}a^{11}-\frac{71}{90}a^{10}-\frac{297}{10}a^{9}-\frac{107}{6}a^{8}+\frac{8573}{30}a^{7}+\frac{19718}{45}a^{6}-\frac{57529}{90}a^{5}-\frac{83731}{45}a^{4}-\frac{60922}{45}a^{3}-\frac{4162}{15}a^{2}+\frac{688}{15}a+\frac{139}{10}$, $\frac{373}{45}a^{11}-\frac{211}{45}a^{10}-\frac{4306}{15}a^{9}-\frac{1613}{6}a^{8}+\frac{242213}{90}a^{7}+\frac{232291}{45}a^{6}-\frac{436583}{90}a^{5}-\frac{304514}{15}a^{4}-\frac{1674113}{90}a^{3}-\frac{169213}{30}a^{2}+\frac{3361}{15}a+\frac{1059}{5}$ | 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: | \( 8958450.47427 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 8958450.47427 \cdot 1}{2\cdot\sqrt{734661949275345164544}}\cr\approx \mathstrut & 0.676891524121 \end{aligned}\]
Galois group
A non-solvable group of order 60 |
The 5 conjugacy class representatives for $A_5$ |
Character table for $A_5$ |
Intermediate fields
6.6.9034882704.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.5.2258720676.1 |
Degree 6 sibling: | 6.6.9034882704.1 |
Degree 10 sibling: | 10.10.20407276368759587904.1 |
Degree 15 sibling: | deg 15 |
Degree 20 sibling: | deg 20 |
Degree 30 sibling: | data not computed |
Minimal sibling: | 5.5.2258720676.1 |
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 | ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{6}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(89\) | 89.2.0.1 | $x^{2} + 82 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
89.10.8.1 | $x^{10} + 410 x^{9} + 67255 x^{8} + 5518600 x^{7} + 226666130 x^{6} + 3740502830 x^{5} + 680034880 x^{4} + 55646420 x^{3} + 491876585 x^{2} + 20083553380 x + 329222720742$ | $5$ | $2$ | $8$ | $D_5$ | $[\ ]_{5}^{2}$ |