Normalized defining polynomial
\( x^{7} - 609x^{5} - 609x^{4} + 104951x^{3} + 119770x^{2} - 5123720x - 9825577 \)
Invariants
Degree: | $7$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[7, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(8233120419813614521\) \(\medspace = 7^{12}\cdot 29^{6}\) | 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: | \(503.76\) | 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: | $7^{12/7}29^{6/7}\approx 503.75978549322235$ | ||
Ramified primes: | \(7\), \(29\) | 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: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $7$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(1421=7^{2}\cdot 29\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1421}(1,·)$, $\chi_{1421}(547,·)$, $\chi_{1421}(372,·)$, $\chi_{1421}(806,·)$, $\chi_{1421}(281,·)$, $\chi_{1421}(799,·)$, $\chi_{1421}(239,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{113822613498089}a^{6}-\frac{9374298642451}{113822613498089}a^{5}+\frac{42193219117896}{113822613498089}a^{4}+\frac{1772490891728}{113822613498089}a^{3}-\frac{5897967116986}{113822613498089}a^{2}+\frac{24146356812414}{113822613498089}a-\frac{45240096311826}{113822613498089}$
Monogenic: | No | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{7}$, which has order $7$
Unit group
Rank: | $6$ | 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: | $\frac{1721235954}{113822613498089}a^{6}-\frac{44453332254}{113822613498089}a^{5}-\frac{820238967584}{113822613498089}a^{4}+\frac{11800695291071}{113822613498089}a^{3}+\frac{83937904363578}{113822613498089}a^{2}-\frac{748567957093335}{113822613498089}a-\frac{17\!\cdots\!76}{113822613498089}$, $\frac{8909741906}{113822613498089}a^{6}-\frac{172918457292}{113822613498089}a^{5}-\frac{2759682195607}{113822613498089}a^{4}+\frac{59834941526453}{113822613498089}a^{3}+\frac{15641398759853}{113822613498089}a^{2}-\frac{32\!\cdots\!20}{113822613498089}a+\frac{70\!\cdots\!92}{113822613498089}$, $\frac{34403866674}{113822613498089}a^{6}+\frac{399222355206}{113822613498089}a^{5}-\frac{16466494326052}{113822613498089}a^{4}-\frac{212281836012246}{113822613498089}a^{3}+\frac{12\!\cdots\!63}{113822613498089}a^{2}+\frac{18\!\cdots\!97}{113822613498089}a+\frac{32\!\cdots\!43}{113822613498089}$, $\frac{11771560021}{113822613498089}a^{6}-\frac{125562362843}{113822613498089}a^{5}-\frac{6508371798152}{113822613498089}a^{4}+\frac{59013145495510}{113822613498089}a^{3}+\frac{919139901471414}{113822613498089}a^{2}-\frac{72\!\cdots\!57}{113822613498089}a-\frac{16\!\cdots\!59}{113822613498089}$, $\frac{9713488503}{113822613498089}a^{6}-\frac{148602812668}{113822613498089}a^{5}-\frac{3399395732878}{113822613498089}a^{4}+\frac{44030612050840}{113822613498089}a^{3}+\frac{247336472253065}{113822613498089}a^{2}-\frac{23\!\cdots\!72}{113822613498089}a-\frac{52\!\cdots\!36}{113822613498089}$, $\frac{27447411872}{113822613498089}a^{6}+\frac{358136103890}{113822613498089}a^{5}-\frac{13531822105327}{113822613498089}a^{4}-\frac{196081860786116}{113822613498089}a^{3}+\frac{10\!\cdots\!03}{113822613498089}a^{2}+\frac{19\!\cdots\!82}{113822613498089}a+\frac{33\!\cdots\!15}{113822613498089}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 1232531.80142 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{0}\cdot 1232531.80142 \cdot 7}{2\cdot\sqrt{8233120419813614521}}\cr\approx \mathstrut & 0.192439364412 \end{aligned}\]
Galois group
A cyclic group of order 7 |
The 7 conjugacy class representatives for $C_7$ |
Character table for $C_7$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.7.0.1}{7} }$ | ${\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.7.0.1}{7} }$ | R | ${\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.7.0.1}{7} }$ | ${\href{/padicField/23.1.0.1}{1} }^{7}$ | R | ${\href{/padicField/31.7.0.1}{7} }$ | ${\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }$ | ${\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.7.0.1}{7} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.7.12.2 | $x^{7} + 42 x^{6} + 56$ | $7$ | $1$ | $12$ | $C_7$ | $[2]$ |
\(29\) | 29.7.6.4 | $x^{7} + 174$ | $7$ | $1$ | $6$ | $C_7$ | $[\ ]_{7}$ |