Normalized defining polynomial
\( x^{12} - x^{11} - 10 x^{10} + 6 x^{9} + 32 x^{8} - 12 x^{7} - 41 x^{6} - 16 x^{5} - 24 x^{4} - 19 x^{3} + \cdots + 1 \)
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)
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Discriminant: | \(2154038935140625\) \(\medspace = 5^{6}\cdot 13^{10}\) | 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: | \(18.96\) | 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: | $5^{1/2}13^{5/6}\approx 18.957066304919827$ | ||
Ramified primes: | \(5\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{1}{2372}a^{11}+\frac{49}{2372}a^{10}+\frac{17}{593}a^{9}+\frac{441}{2372}a^{8}-\frac{113}{593}a^{7}-\frac{39}{1186}a^{6}-\frac{244}{593}a^{5}-\frac{783}{2372}a^{4}+\frac{557}{2372}a^{3}+\frac{573}{1186}a^{2}-\frac{805}{2372}a+\frac{169}{593}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$
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: | $\frac{95}{1186}a^{11}-\frac{89}{1186}a^{10}-\frac{1905}{2372}a^{9}+\frac{385}{1186}a^{8}+\frac{1657}{593}a^{7}+\frac{5}{2372}a^{6}-\frac{11691}{2372}a^{5}-\frac{7043}{2372}a^{4}+\frac{3241}{2372}a^{3}-\frac{2021}{1186}a^{2}+\frac{3009}{2372}a+\frac{5689}{2372}$, $\frac{3}{593}a^{11}-\frac{5}{2372}a^{10}-\frac{185}{1186}a^{9}+\frac{137}{593}a^{8}+\frac{2285}{2372}a^{7}-\frac{3901}{2372}a^{6}-\frac{4003}{2372}a^{5}+\frac{7801}{2372}a^{4}-\frac{108}{593}a^{3}+\frac{1299}{2372}a^{2}+\frac{3979}{2372}a-\frac{95}{1186}$, $\frac{3}{593}a^{11}-\frac{5}{2372}a^{10}-\frac{185}{1186}a^{9}+\frac{137}{593}a^{8}+\frac{2285}{2372}a^{7}-\frac{3901}{2372}a^{6}-\frac{4003}{2372}a^{5}+\frac{7801}{2372}a^{4}-\frac{108}{593}a^{3}+\frac{1299}{2372}a^{2}+\frac{3979}{2372}a-\frac{1281}{1186}$, $\frac{379}{1186}a^{11}-\frac{499}{593}a^{10}-\frac{1346}{593}a^{9}+\frac{3811}{593}a^{8}+\frac{2110}{593}a^{7}-\frac{18295}{1186}a^{6}+\frac{3093}{1186}a^{5}+\frac{2540}{593}a^{4}-\frac{9493}{1186}a^{3}+\frac{7967}{1186}a^{2}+\frac{5637}{1186}a-\frac{2937}{1186}$, $\frac{1400}{593}a^{11}-\frac{2560}{593}a^{10}-\frac{47939}{2372}a^{9}+\frac{18470}{593}a^{8}+\frac{60941}{1186}a^{7}-\frac{171729}{2372}a^{6}-\frac{98357}{2372}a^{5}-\frac{751}{2372}a^{4}-\frac{122731}{2372}a^{3}+\frac{77}{1186}a^{2}+\frac{63437}{2372}a+\frac{5225}{2372}$, $\frac{4823}{2372}a^{11}-\frac{7989}{2372}a^{10}-\frac{42661}{2372}a^{9}+\frac{56187}{2372}a^{8}+\frac{57457}{1186}a^{7}-\frac{127727}{2372}a^{6}-\frac{108533}{2372}a^{5}-\frac{4642}{593}a^{4}-\frac{26210}{593}a^{3}-\frac{10475}{1186}a^{2}+\frac{25427}{1186}a+\frac{13673}{2372}$, $\frac{499}{593}a^{11}-\frac{1048}{593}a^{10}-\frac{4020}{593}a^{9}+\frac{7765}{593}a^{8}+\frac{17967}{1186}a^{7}-\frac{36927}{1186}a^{6}-\frac{8051}{1186}a^{5}+\frac{5477}{1186}a^{4}-\frac{24661}{1186}a^{3}+\frac{6927}{1186}a^{2}+\frac{9613}{1186}a-\frac{1279}{593}$ | 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: | \( 861.443546399 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 861.443546399 \cdot 2}{2\cdot\sqrt{2154038935140625}}\cr\approx \mathstrut & 0.462849184982 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 12T7):
A solvable group of order 24 |
The 8 conjugacy class representatives for $A_4 \times C_2$ |
Character table for $A_4 \times C_2$ |
Intermediate fields
\(\Q(\sqrt{65}) \), 3.3.169.1, 6.6.46411625.1, 6.2.1856465.1, 6.2.714025.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 6 sibling: | 6.2.1856465.1 |
Degree 8 sibling: | 8.0.3016755625.3 |
Degree 12 sibling: | 12.0.86161557405625.3 |
Minimal sibling: | 6.2.1856465.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{4}$ | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(13\) | 13.6.5.5 | $x^{6} + 65$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
13.6.5.5 | $x^{6} + 65$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |