Normalized defining polynomial
\( x^{14} - 50x^{12} + 1800x^{10} - 34000x^{8} + 440000x^{6} - 2400000x^{4} + 1000000x^{2} + 17500000 \)
Invariants
Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-11018777412256797818880000000\) \(\medspace = -\,2^{27}\cdot 3^{12}\cdot 5^{7}\cdot 7^{11}\) | 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: | \(100.70\) | 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: | $2^{27/14}3^{6/7}5^{1/2}7^{5/6}\approx 110.4720194582544$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-70}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $\frac{1}{5}a^{2}$, $\frac{1}{10}a^{3}$, $\frac{1}{50}a^{4}$, $\frac{1}{50}a^{5}$, $\frac{1}{500}a^{6}$, $\frac{1}{500}a^{7}$, $\frac{1}{2500}a^{8}$, $\frac{1}{5000}a^{9}$, $\frac{1}{325000}a^{10}-\frac{3}{16250}a^{8}-\frac{3}{6500}a^{6}-\frac{3}{325}a^{4}-\frac{1}{13}a^{2}-\frac{1}{13}$, $\frac{1}{325000}a^{11}+\frac{1}{65000}a^{9}-\frac{3}{6500}a^{7}-\frac{3}{325}a^{5}+\frac{3}{130}a^{3}-\frac{1}{13}a$, $\frac{1}{3250000}a^{12}-\frac{1}{6500}a^{8}+\frac{1}{3250}a^{6}-\frac{1}{325}a^{4}+\frac{2}{65}a^{2}-\frac{6}{13}$, $\frac{1}{3250000}a^{13}+\frac{3}{65000}a^{9}+\frac{1}{3250}a^{7}-\frac{1}{325}a^{5}+\frac{2}{65}a^{3}-\frac{6}{13}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)
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)
| |
Fundamental units: | $\frac{19}{3250000}a^{12}-\frac{21}{81250}a^{10}+\frac{73}{8125}a^{8}-\frac{189}{1300}a^{6}+\frac{1103}{650}a^{4}-\frac{257}{65}a^{2}-\frac{225}{13}$, $\frac{21}{650000}a^{12}-\frac{463}{325000}a^{10}+\frac{404}{8125}a^{8}-\frac{201}{250}a^{6}+\frac{6143}{650}a^{4}-\frac{1401}{65}a^{2}-\frac{1233}{13}$, $\frac{63}{3250000}a^{12}-\frac{11}{13000}a^{10}+\frac{479}{16250}a^{8}-\frac{1533}{3250}a^{6}+\frac{3591}{650}a^{4}-\frac{774}{65}a^{2}-\frac{701}{13}$, $\frac{12443}{650000}a^{13}+\frac{76731}{1625000}a^{12}-\frac{136613}{162500}a^{11}-\frac{673971}{325000}a^{10}+\frac{1907281}{65000}a^{9}+\frac{2352359}{32500}a^{8}-\frac{3070253}{6500}a^{7}-\frac{1893449}{1625}a^{6}+\frac{3606847}{650}a^{5}+\frac{684427}{50}a^{4}-\frac{316701}{26}a^{3}-\frac{390803}{13}a^{2}-\frac{715471}{13}a-\frac{1765331}{13}$, $\frac{1382451}{3250000}a^{13}+\frac{1096579}{325000}a^{12}-\frac{579021}{162500}a^{11}-\frac{32014763}{325000}a^{10}+\frac{75589}{1000}a^{9}+\frac{17528997}{6500}a^{8}+\frac{6342451}{1300}a^{7}-\frac{30682464}{1625}a^{6}-\frac{7067083}{130}a^{5}+\frac{789579}{650}a^{4}+\frac{1699875}{26}a^{3}+\frac{857593}{5}a^{2}+\frac{5467837}{13}a+\frac{1601473}{13}$, $\frac{34\!\cdots\!21}{812500}a^{13}+\frac{12\!\cdots\!77}{3250000}a^{12}-\frac{853373488307001}{65000}a^{11}-\frac{17\!\cdots\!03}{162500}a^{10}-\frac{30\!\cdots\!21}{32500}a^{9}+\frac{67\!\cdots\!67}{32500}a^{8}+\frac{33\!\cdots\!07}{6500}a^{7}-\frac{41\!\cdots\!49}{3250}a^{6}-\frac{14\!\cdots\!47}{325}a^{5}-\frac{84\!\cdots\!47}{650}a^{4}+\frac{54\!\cdots\!47}{13}a^{3}+\frac{14\!\cdots\!59}{13}a^{2}+\frac{43\!\cdots\!35}{13}a+\frac{10\!\cdots\!39}{13}$ (assuming GRH) | 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: | \( 96837594.26008515 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 96837594.26008515 \cdot 4}{2\cdot\sqrt{11018777412256797818880000000}}\cr\approx \mathstrut & 0.713290370185430 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times F_7$ (as 14T7):
A solvable group of order 84 |
The 14 conjugacy class representatives for $F_7 \times C_2$ |
Character table for $F_7 \times C_2$ |
Intermediate fields
\(\Q(\sqrt{-70}) \), 7.1.784147392.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 14 sibling: | data not computed |
Degree 28 sibling: | data not computed |
Degree 42 siblings: | data not computed |
Minimal sibling: | 14.2.1574111058893828259840000000.1 |
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 | R | R | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.14.0.1}{14} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.14.27.13 | $x^{14} + 4 x^{12} + 4 x^{11} + 4 x^{9} + 4 x^{8} + 4 x^{6} + 4 x^{4} + 6$ | $14$ | $1$ | $27$ | $(C_7:C_3) \times C_2$ | $[3]_{7}^{3}$ |
\(3\) | 3.14.12.1 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
\(7\) | 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |