Normalized defining polynomial
\( x^{12} - x^{11} - 7x^{10} + 2x^{9} + 7x^{8} - 2x^{7} - 127x^{6} - 77x^{5} + 98x^{4} + 28x^{3} + 49 \)
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: | \(2572632521265625\) \(\medspace = 5^{6}\cdot 7^{8}\cdot 13^{4}\) | 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: | \(19.24\) | 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: | $5^{1/2}7^{2/3}13^{1/2}\approx 29.50226581402583$ | ||
Ramified primes: | \(5\), \(7\), \(13\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7}a^{7}-\frac{3}{7}a^{6}+\frac{3}{7}a^{5}-\frac{2}{7}a^{4}+\frac{2}{7}a^{3}$, $\frac{1}{14}a^{8}-\frac{1}{14}a^{7}+\frac{2}{7}a^{6}-\frac{3}{14}a^{5}-\frac{1}{7}a^{4}+\frac{2}{7}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{14}a^{9}-\frac{1}{14}a^{7}-\frac{1}{14}a^{6}-\frac{3}{14}a^{5}-\frac{2}{7}a^{4}+\frac{3}{14}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{28}a^{10}-\frac{1}{14}a^{7}+\frac{1}{28}a^{6}-\frac{1}{4}a^{5}+\frac{1}{28}a^{4}+\frac{11}{28}a^{3}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{2702666372}a^{11}+\frac{41240009}{2702666372}a^{10}-\frac{23815315}{1351333186}a^{9}+\frac{7535152}{675666593}a^{8}-\frac{145723469}{2702666372}a^{7}+\frac{203829755}{675666593}a^{6}-\frac{279060619}{1351333186}a^{5}-\frac{143968101}{675666593}a^{4}+\frac{238961077}{2702666372}a^{3}-\frac{26685249}{386095196}a^{2}+\frac{33933844}{96523799}a+\frac{132903603}{386095196}$
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: | $\frac{243114}{675666593}a^{11}+\frac{22097}{675666593}a^{10}-\frac{2387187}{675666593}a^{9}-\frac{427773}{675666593}a^{8}+\frac{4075901}{675666593}a^{7}-\frac{1995740}{96523799}a^{6}-\frac{11020269}{675666593}a^{5}+\frac{11062097}{675666593}a^{4}+\frac{4369648}{675666593}a^{3}-\frac{46998}{96523799}a^{2}-\frac{106060}{96523799}a+\frac{60491085}{96523799}$, $\frac{194569}{65918692}a^{11}-\frac{115461}{9416956}a^{10}-\frac{529441}{32959346}a^{9}+\frac{2241759}{32959346}a^{8}+\frac{3048161}{65918692}a^{7}-\frac{638762}{16479673}a^{6}-\frac{6739158}{16479673}a^{5}+\frac{16147062}{16479673}a^{4}+\frac{14252275}{9416956}a^{3}+\frac{5939001}{9416956}a^{2}-\frac{953459}{2354239}a-\frac{7084547}{9416956}$, $\frac{110716}{16479673}a^{11}+\frac{7894}{16479673}a^{10}-\frac{1013470}{16479673}a^{9}-\frac{594897}{32959346}a^{8}+\frac{3253639}{32959346}a^{7}-\frac{114949}{2354239}a^{6}-\frac{4163431}{4708478}a^{5}-\frac{23355589}{16479673}a^{4}+\frac{2231319}{2354239}a^{3}-\frac{26015}{4708478}a^{2}-\frac{1948491}{2354239}a+\frac{147493}{4708478}$, $\frac{27107209}{2702666372}a^{11}-\frac{4536329}{386095196}a^{10}-\frac{109585995}{1351333186}a^{9}+\frac{33332898}{675666593}a^{8}+\frac{408076603}{2702666372}a^{7}-\frac{73149785}{675666593}a^{6}-\frac{1769556191}{1351333186}a^{5}-\frac{284487510}{675666593}a^{4}+\frac{6669435729}{2702666372}a^{3}+\frac{241177819}{386095196}a^{2}-\frac{119086010}{96523799}a+\frac{349687535}{386095196}$, $\frac{8949785}{2702666372}a^{11}-\frac{33048919}{2702666372}a^{10}-\frac{3783065}{193047598}a^{9}+\frac{91056573}{1351333186}a^{8}+\frac{141278205}{2702666372}a^{7}-\frac{40159422}{675666593}a^{6}-\frac{287325747}{675666593}a^{5}+\frac{673091639}{675666593}a^{4}+\frac{4107881517}{2702666372}a^{3}+\frac{243311049}{386095196}a^{2}-\frac{39197879}{96523799}a+\frac{337593109}{386095196}$, $\frac{4804567}{675666593}a^{11}-\frac{361735}{675666593}a^{10}-\frac{42466271}{675666593}a^{9}-\frac{19642571}{1351333186}a^{8}+\frac{106431295}{1351333186}a^{7}-\frac{13135154}{675666593}a^{6}-\frac{1135340947}{1351333186}a^{5}-\frac{977034673}{675666593}a^{4}+\frac{90464625}{96523799}a^{3}-\frac{1278735}{193047598}a^{2}+\frac{77126753}{96523799}a+\frac{2643617}{193047598}$, $\frac{18790101}{1351333186}a^{11}+\frac{34103043}{2702666372}a^{10}-\frac{143733991}{1351333186}a^{9}-\frac{107055988}{675666593}a^{8}+\frac{7350675}{193047598}a^{7}+\frac{27010941}{386095196}a^{6}-\frac{5001652585}{2702666372}a^{5}-\frac{11971446815}{2702666372}a^{4}-\frac{7426582789}{2702666372}a^{3}-\frac{35603957}{96523799}a^{2}-\frac{724320755}{386095196}a-\frac{678995813}{386095196}$ | 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: | \( 709.275411867 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 709.275411867 \cdot 1}{2\cdot\sqrt{2572632521265625}}\cr\approx \mathstrut & 0.174355434049 \end{aligned}\]
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
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 6.6.300125.1, 6.2.50721125.1, 6.2.405769.1 |
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: | 6.2.50721125.1 |
Degree 8 sibling: | 8.0.42859350625.2 |
Degree 12 sibling: | 12.0.434774896093890625.1 |
Minimal sibling: | 6.2.50721125.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }^{2}$ | R | R | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(7\) | 7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
7.6.4.3 | $x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |