Normalized defining polynomial
\( x^{12} - 2x^{11} - 2x^{10} + 7x^{9} - 10x^{8} + 11x^{7} - 2x^{6} - 5x^{5} + 4x^{4} + 5x^{3} - 14x^{2} + 7x - 1 \)
Invariants
Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(53697370966801\) \(\medspace = 2707^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.94\) | 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: | $2707^{1/2}\approx 52.028838157314254$ | ||
Ramified primes: | \(2707\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{949}a^{11}-\frac{62}{949}a^{10}-\frac{6}{73}a^{9}-\frac{58}{949}a^{8}-\frac{326}{949}a^{7}-\frac{358}{949}a^{6}-\frac{349}{949}a^{5}+\frac{57}{949}a^{4}+\frac{380}{949}a^{3}-\frac{19}{949}a^{2}+\frac{177}{949}a-\frac{174}{949}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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)
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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: | $a$, $\frac{2086}{949}a^{11}+\frac{4064}{949}a^{10}+\frac{398}{73}a^{9}-\frac{14719}{949}a^{8}+\frac{17634}{949}a^{7}-\frac{20004}{949}a^{6}-\frac{818}{949}a^{5}+\frac{12060}{949}a^{4}-\frac{5010}{949}a^{3}-\frac{11612}{949}a^{2}+\frac{29358}{949}a-\frac{7146}{949}$, $\frac{5060}{949}a^{11}+\frac{8142}{949}a^{10}+\frac{1014}{73}a^{9}-\frac{30129}{949}a^{8}+\frac{39107}{949}a^{7}-\frac{40968}{949}a^{6}-\frac{4894}{949}a^{5}+\frac{22852}{949}a^{4}-\frac{11514}{949}a^{3}-\frac{29128}{949}a^{2}+\frac{59074}{949}a-\frac{14467}{949}$, $\frac{5060}{949}a^{11}+\frac{8142}{949}a^{10}+\frac{1014}{73}a^{9}-\frac{30129}{949}a^{8}+\frac{39107}{949}a^{7}-\frac{40968}{949}a^{6}-\frac{4894}{949}a^{5}+\frac{22852}{949}a^{4}-\frac{11514}{949}a^{3}-\frac{29128}{949}a^{2}+\frac{59074}{949}a-\frac{13518}{949}$, $\frac{4905}{949}a^{11}-\frac{8022}{949}a^{10}-\frac{960}{73}a^{9}+\frac{29629}{949}a^{8}-\frac{38874}{949}a^{7}+\frac{40467}{949}a^{6}+\frac{2998}{949}a^{5}-\frac{22197}{949}a^{4}+\frac{12401}{949}a^{3}+\frac{28277}{949}a^{2}-\frac{58039}{949}a+\frac{15814}{949}$, $\frac{2025}{949}a^{11}+\frac{3129}{949}a^{10}+\frac{397}{73}a^{9}-\frac{11614}{949}a^{8}+\frac{15779}{949}a^{7}-\frac{16219}{949}a^{6}-\frac{1229}{949}a^{5}+\frac{8894}{949}a^{4}-\frac{4606}{949}a^{3}-\frac{11822}{949}a^{2}+\frac{23073}{949}a-\frac{6372}{949}$, $\frac{2025}{949}a^{11}-\frac{3129}{949}a^{10}-\frac{397}{73}a^{9}+\frac{11614}{949}a^{8}-\frac{15779}{949}a^{7}+\frac{16219}{949}a^{6}+\frac{1229}{949}a^{5}-\frac{8894}{949}a^{4}+\frac{4606}{949}a^{3}+\frac{11822}{949}a^{2}-\frac{22124}{949}a+\frac{6372}{949}$ | 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: | \( 138.27818534649447 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 138.27818534649447 \cdot 1}{2\cdot\sqrt{53697370966801}}\cr\approx \mathstrut & 0.235280871656535 \end{aligned}\]
Galois group
$C_2\times A_5$ (as 12T76):
A non-solvable group of order 120 |
The 10 conjugacy class representatives for $C_2\times A_5$ |
Character table for $C_2\times A_5$ |
Intermediate fields
6.2.7327849.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 10 sibling: | data not computed |
Degree 12 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 sibling: | data not computed |
Minimal sibling: | 10.0.145358783207130307.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.2.0.1}{2} }^{6}$ | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2707\) | $\Q_{2707}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2707}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{2707}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{2707}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |