Normalized defining polynomial
\( x^{42} + 41 x^{40} + 780 x^{38} + 9139 x^{36} + 73815 x^{34} + 435897 x^{32} + 1947792 x^{30} + 6724520 x^{28} + \cdots + 1 \)
Invariants
Degree: | $42$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 21]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-959396304051793463814262846982490027578741814649477038563926538598268329263104\) \(\medspace = -\,2^{42}\cdot 43^{40}\) | 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: | \(71.90\) | 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: | $2\cdot 43^{20/21}\approx 71.89757443060533$ | ||
Ramified primes: | \(2\), \(43\) | 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: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Gal(K/\Q) }$: | $42$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(172=2^{2}\cdot 43\) | ||
Dirichlet character group: | $\lbrace$$\chi_{172}(1,·)$, $\chi_{172}(17,·)$, $\chi_{172}(133,·)$, $\chi_{172}(135,·)$, $\chi_{172}(9,·)$, $\chi_{172}(11,·)$, $\chi_{172}(13,·)$, $\chi_{172}(15,·)$, $\chi_{172}(145,·)$, $\chi_{172}(21,·)$, $\chi_{172}(23,·)$, $\chi_{172}(25,·)$, $\chi_{172}(31,·)$, $\chi_{172}(35,·)$, $\chi_{172}(165,·)$, $\chi_{172}(167,·)$, $\chi_{172}(41,·)$, $\chi_{172}(47,·)$, $\chi_{172}(49,·)$, $\chi_{172}(53,·)$, $\chi_{172}(57,·)$, $\chi_{172}(59,·)$, $\chi_{172}(67,·)$, $\chi_{172}(79,·)$, $\chi_{172}(81,·)$, $\chi_{172}(83,·)$, $\chi_{172}(139,·)$, $\chi_{172}(87,·)$, $\chi_{172}(143,·)$, $\chi_{172}(95,·)$, $\chi_{172}(97,·)$, $\chi_{172}(99,·)$, $\chi_{172}(101,·)$, $\chi_{172}(103,·)$, $\chi_{172}(107,·)$, $\chi_{172}(109,·)$, $\chi_{172}(111,·)$, $\chi_{172}(117,·)$, $\chi_{172}(169,·)$, $\chi_{172}(153,·)$, $\chi_{172}(121,·)$, $\chi_{172}(127,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{1048576}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{1878326}$, which has order $3756652$ (assuming GRH)
Unit group
Rank: | $20$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -a^{41} - 40 a^{39} - 741 a^{37} - 8436 a^{35} - 66045 a^{33} - 376992 a^{31} - 1623160 a^{29} - 5379616 a^{27} - 13884156 a^{25} - 28048800 a^{23} - 44352165 a^{21} - 54627300 a^{19} - 51895935 a^{17} - 37442160 a^{15} - 20058300 a^{13} - 7726160 a^{11} - 2042975 a^{9} - 346104 a^{7} - 33649 a^{5} - 1540 a^{3} - 21 a \) (order $4$) | 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: | $a^{39}+39a^{37}+702a^{35}+7735a^{33}+58344a^{31}+319176a^{29}+1308944a^{27}+4102137a^{25}+9924525a^{23}+18599295a^{21}+26936910a^{19}+29910465a^{17}+25110020a^{15}+15600900a^{13}+6953544a^{11}+2124694a^{9}+415701a^{7}+46683a^{5}+2470a^{3}+39a$, $a^{30}+30a^{28}+405a^{26}+3250a^{24}+17250a^{22}+63756a^{20}+168245a^{18}+319770a^{16}+436050a^{14}+419900a^{12}+277134a^{10}+119340a^{8}+30940a^{6}+4200a^{4}+225a^{2}+2$, $a^{26}+26a^{24}+299a^{22}+2002a^{20}+8645a^{18}+25194a^{16}+50388a^{14}+68952a^{12}+63206a^{10}+37180a^{8}+13013a^{6}+2366a^{4}+169a^{2}+1$, $a^{28}+28a^{26}+350a^{24}+2576a^{22}+12397a^{20}+40964a^{18}+94962a^{16}+155040a^{14}+176358a^{12}+136136a^{10}+68068a^{8}+20384a^{6}+3185a^{4}+196a^{2}+2$, $a^{2}+2$, $a^{26}+26a^{24}+299a^{22}+2002a^{20}+8645a^{18}+25194a^{16}+50388a^{14}+68952a^{12}+63206a^{10}+37180a^{8}+13013a^{6}+2366a^{4}+169a^{2}+2$, $a^{35}+35a^{33}+560a^{31}+5425a^{29}+35525a^{27}+166257a^{25}+573300a^{23}+1480050a^{21}+2877875a^{19}+4206125a^{17}+4576264a^{15}+3640210a^{13}+2057510a^{11}+791350a^{9}+193800a^{7}+27132a^{5}+1785a^{3}+35a$, $a^{34}+34a^{32}+527a^{30}+4930a^{28}+31059a^{26}+139230a^{24}+457470a^{22}+1118260a^{20}+2042975a^{18}+2778446a^{16}+2778446a^{14}+1998724a^{12}+999362a^{10}+329460a^{8}+65892a^{6}+6936a^{4}+289a^{2}+2$, $a$, $a^{34}+34a^{32}+527a^{30}+4930a^{28}+31060a^{26}+139256a^{24}+457769a^{22}+1120262a^{20}+2051620a^{18}+2803640a^{16}+2828834a^{14}+2067676a^{12}+1062568a^{10}+366640a^{8}+78905a^{6}+9302a^{4}+458a^{2}+3$, $a^{41}+40a^{39}+741a^{37}+8436a^{35}+66045a^{33}+376992a^{31}+1623160a^{29}+5379617a^{27}+13884183a^{25}+28049124a^{23}+44354442a^{21}+54637695a^{19}+51928254a^{17}+37511928a^{15}+20162952a^{13}+7833567a^{11}+2115916a^{9}+377036a^{7}+41097a^{5}+2414a^{3}+59a$, $a^{24}+24a^{22}+252a^{20}+1520a^{18}+5814a^{16}+14688a^{14}+24753a^{12}+27468a^{10}+19359a^{8}+8120a^{6}+1821a^{4}+180a^{2}+5$, $a^{40}+39a^{38}+703a^{36}+7770a^{34}+58905a^{32}+324632a^{30}+1344903a^{28}+4272021a^{26}+10517975a^{24}+20157774a^{22}+30034368a^{20}+34563451a^{18}+30346173a^{16}+19938960a^{14}+9525542a^{12}+3168375a^{10}+685287a^{8}+85596a^{6}+4816a^{4}+57a^{2}+1$, $a^{41}+40a^{39}+741a^{37}+8436a^{35}+66045a^{33}+376992a^{31}+1623160a^{29}+5379616a^{27}+13884156a^{25}+28048801a^{23}+44352188a^{21}+54627530a^{19}+51897246a^{17}+37446852a^{15}+20069248a^{13}+7742904a^{11}+2059420a^{9}+355971a^{7}+36938a^{5}+2046a^{3}+44a$, $a^{38}+38a^{36}+665a^{34}+7106a^{32}+51832a^{30}+273295a^{28}+1076075a^{26}+3223000a^{24}+7411129a^{22}+13110713a^{20}+17768972a^{18}+18254686a^{16}+13960195a^{14}+7728644a^{12}+2970473a^{10}+743798a^{8}+109879a^{6}+8185a^{4}+246a^{2}+2$, $a^{16}+16a^{14}+104a^{12}+352a^{10}+660a^{8}+672a^{6}+336a^{4}+64a^{2}+3$, $a^{14}+14a^{12}+77a^{10}+210a^{8}+294a^{6}+196a^{4}+49a^{2}+2$, $a^{28}+28a^{26}+350a^{24}+2576a^{22}+12397a^{20}+40964a^{18}+94962a^{16}+155039a^{14}+176344a^{12}+136059a^{10}+67858a^{8}+20090a^{6}+2989a^{4}+147a^{2}+1$, $a^{41}+40a^{39}+741a^{37}+8437a^{35}+66080a^{33}+377552a^{31}+1628585a^{29}+5415141a^{27}+14050413a^{25}+28622100a^{23}+45832215a^{21}+57505175a^{19}+56102060a^{17}+42018424a^{15}+23698510a^{13}+9783670a^{11}+2834325a^{9}+539904a^{7}+60781a^{5}+3325a^{3}+56a$, $a^{35}+35a^{33}+560a^{31}+5425a^{29}+35525a^{27}+166257a^{25}+573300a^{23}+1480050a^{21}+2877876a^{19}+4206144a^{17}+4576416a^{15}+3640875a^{13}+2059239a^{11}+794067a^{9}+196308a^{7}+28386a^{5}+2070a^{3}+54a$ (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)]
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Regulator: | \( 2748021948787771.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{21}\cdot 2748021948787771.5 \cdot 3756652}{4\cdot\sqrt{959396304051793463814262846982490027578741814649477038563926538598268329263104}}\cr\approx \mathstrut & 0.152243719791663 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 42 |
The 42 conjugacy class representatives for $C_{42}$ |
Character table for $C_{42}$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), 3.3.1849.1, 6.0.218803264.1, 7.7.6321363049.1, 14.0.654698590982350051753984.1, \(\Q(\zeta_{43})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | $42$ | $21^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{7}$ | ${\href{/padicField/11.14.0.1}{14} }^{3}$ | $21^{2}$ | $21^{2}$ | $42$ | $42$ | $21^{2}$ | $42$ | ${\href{/padicField/37.3.0.1}{3} }^{14}$ | ${\href{/padicField/41.7.0.1}{7} }^{6}$ | R | ${\href{/padicField/47.14.0.1}{14} }^{3}$ | $21^{2}$ | ${\href{/padicField/59.14.0.1}{14} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.14.14.38 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1948 x^{10} + 8392 x^{9} + 30520 x^{8} + 84992 x^{7} + 178608 x^{6} + 284064 x^{5} + 325984 x^{4} + 242688 x^{3} + 97600 x^{2} + 11648 x - 5504$ | $2$ | $7$ | $14$ | $C_{14}$ | $[2]^{7}$ |
2.14.14.38 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1948 x^{10} + 8392 x^{9} + 30520 x^{8} + 84992 x^{7} + 178608 x^{6} + 284064 x^{5} + 325984 x^{4} + 242688 x^{3} + 97600 x^{2} + 11648 x - 5504$ | $2$ | $7$ | $14$ | $C_{14}$ | $[2]^{7}$ | |
2.14.14.38 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1948 x^{10} + 8392 x^{9} + 30520 x^{8} + 84992 x^{7} + 178608 x^{6} + 284064 x^{5} + 325984 x^{4} + 242688 x^{3} + 97600 x^{2} + 11648 x - 5504$ | $2$ | $7$ | $14$ | $C_{14}$ | $[2]^{7}$ | |
\(43\) | Deg $42$ | $21$ | $2$ | $40$ |