LMFDB - Number field 20.4.4800653894273660658369343520768.2

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Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^18 + 10*x^16 - 120*x^14 + 210*x^12 - 270*x^10 + 1100*x^8 - 728*x^6 + 193*x^4 - 23*x^2 + 2)

gp:K = bnfinit(y^20 - 7*y^18 + 10*y^16 - 120*y^14 + 210*y^12 - 270*y^10 + 1100*y^8 - 728*y^6 + 193*y^4 - 23*y^2 + 2, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 7*x^18 + 10*x^16 - 120*x^14 + 210*x^12 - 270*x^10 + 1100*x^8 - 728*x^6 + 193*x^4 - 23*x^2 + 2);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 7*x^18 + 10*x^16 - 120*x^14 + 210*x^12 - 270*x^10 + 1100*x^8 - 728*x^6 + 193*x^4 - 23*x^2 + 2)

$ x^{20} - 7x^{18} + 10x^{16} - 120x^{14} + 210x^{12} - 270x^{10} + 1100x^{8} - 728x^{6} + 193x^{4} - 23x^{2} + 2 $ x^20 - 7*x^18 + 10*x^16 - 120*x^14 + 210*x^12 - 270*x^10 + 1100*x^8 - 728*x^6 + 193*x^4 - 23*x^2 + 2

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$20$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$(4, 8)$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$4800653894273660658369343520768$ 4800653894273660658369343520768 $\medspace = 2^{61}\cdot 113^{6}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$34.20$	comment:Root discriminant sage:(K.disc().abs())^(1./K.degree()) gp:abs(K.disc)^(1/poldegree(K.pol)) magma:Abs(Discriminant(OK))^(1/Degree(K)); oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
Galois root discriminant:	not computed
Ramified primes:	$2$, $113$ 2, 113	comment:Ramified primes sage:K.disc().support() gp:factor(abs(K.disc))[,1]~ magma:PrimeDivisors(Discriminant(OK)); oscar:prime_divisors(discriminant(OK))
Discriminant root field:	$\Q(\sqrt{2}) $
$\Aut(K/\Q)$:	$C_2$	comment:Automorphisms sage:K.automorphisms() magma:Automorphisms(K); oscar:automorphism_group(K)
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{12}+\frac{1}{8}a^{10}+\frac{1}{4}a^{8}-\frac{3}{8}a^{4}-\frac{3}{8}a^{2}+\frac{1}{4}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{3}{8}a^{5}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{14}+\frac{1}{8}a^{10}-\frac{1}{4}a^{8}-\frac{3}{8}a^{6}-\frac{3}{8}a^{2}-\frac{1}{4}$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{11}-\frac{1}{4}a^{10}+\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{3}{8}a^{7}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{16}+\frac{1}{8}a^{10}+\frac{3}{8}a^{8}-\frac{3}{8}a^{2}-\frac{1}{4}$, $\frac{1}{16}a^{17}-\frac{1}{16}a^{16}-\frac{1}{16}a^{15}-\frac{1}{16}a^{14}-\frac{1}{16}a^{13}-\frac{1}{16}a^{12}-\frac{1}{16}a^{11}-\frac{3}{16}a^{10}+\frac{3}{16}a^{9}+\frac{5}{16}a^{8}+\frac{3}{16}a^{7}-\frac{5}{16}a^{6}+\frac{3}{16}a^{5}-\frac{5}{16}a^{4}+\frac{3}{16}a^{3}+\frac{1}{16}a^{2}-\frac{1}{8}a+\frac{1}{8}$, $\frac{1}{2373159024}a^{18}+\frac{13440925}{593289756}a^{16}-\frac{1623875}{65921084}a^{14}-\frac{3211079}{98881626}a^{12}+\frac{86232433}{395526504}a^{10}-\frac{1}{2}a^{9}-\frac{75754717}{395526504}a^{8}-\frac{1}{2}a^{7}+\frac{27379343}{593289756}a^{6}-\frac{1}{2}a^{5}+\frac{13307936}{148322439}a^{4}-\frac{1}{2}a^{3}-\frac{29300227}{2373159024}a^{2}-\frac{494109545}{1186579512}$, $\frac{1}{2373159024}a^{19}+\frac{13440925}{593289756}a^{17}-\frac{1623875}{65921084}a^{15}-\frac{3211079}{98881626}a^{13}-\frac{12649193}{395526504}a^{11}-\frac{1}{4}a^{10}+\frac{122008535}{395526504}a^{9}+\frac{27379343}{593289756}a^{7}-\frac{1}{2}a^{6}+\frac{13307936}{148322439}a^{5}-\frac{1}{2}a^{4}-\frac{622589983}{2373159024}a^{3}+\frac{1}{4}a^{2}+\frac{99180211}{1186579512}a-\frac{1}{2}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, 1/2*a^10 - 1/2*a^2, 1/4*a^11 - 1/4*a^10 - 1/2*a^9 - 1/2*a^8 + 1/4*a^3 - 1/4*a^2 - 1/2*a - 1/2, 1/8*a^12 + 1/8*a^10 + 1/4*a^8 - 3/8*a^4 - 3/8*a^2 + 1/4, 1/8*a^13 - 1/8*a^11 - 1/4*a^10 - 1/4*a^9 - 1/2*a^8 - 3/8*a^5 + 3/8*a^3 - 1/4*a^2 - 1/4*a - 1/2, 1/8*a^14 + 1/8*a^10 - 1/4*a^8 - 3/8*a^6 - 3/8*a^2 - 1/4, 1/8*a^15 - 1/8*a^11 - 1/4*a^10 + 1/4*a^9 - 1/2*a^8 - 3/8*a^7 + 3/8*a^3 - 1/4*a^2 + 1/4*a - 1/2, 1/8*a^16 + 1/8*a^10 + 3/8*a^8 - 3/8*a^2 - 1/4, 1/16*a^17 - 1/16*a^16 - 1/16*a^15 - 1/16*a^14 - 1/16*a^13 - 1/16*a^12 - 1/16*a^11 - 3/16*a^10 + 3/16*a^9 + 5/16*a^8 + 3/16*a^7 - 5/16*a^6 + 3/16*a^5 - 5/16*a^4 + 3/16*a^3 + 1/16*a^2 - 1/8*a + 1/8, 1/2373159024*a^18 + 13440925/593289756*a^16 - 1623875/65921084*a^14 - 3211079/98881626*a^12 + 86232433/395526504*a^10 - 1/2*a^9 - 75754717/395526504*a^8 - 1/2*a^7 + 27379343/593289756*a^6 - 1/2*a^5 + 13307936/148322439*a^4 - 1/2*a^3 - 29300227/2373159024*a^2 - 494109545/1186579512, 1/2373159024*a^19 + 13440925/593289756*a^17 - 1623875/65921084*a^15 - 3211079/98881626*a^13 - 12649193/395526504*a^11 - 1/4*a^10 + 122008535/395526504*a^9 + 27379343/593289756*a^7 - 1/2*a^6 + 13307936/148322439*a^5 - 1/2*a^4 - 622589983/2373159024*a^3 + 1/4*a^2 + 99180211/1186579512*a - 1/2

comment:Integral basis

sage:K.integral_basis()

gp:K.zk

magma:IntegralBasis(K);

oscar:basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

Ideal class group:		Trivial group, which has order $1$ (assuming GRH)	comment:Class group sage:K.class_group().invariants() gp:K.clgp magma:ClassGroup(K); oscar:class_group(K)
Narrow class group:		$C_{2}$, which has order $2$ (assuming GRH)	comment:Narrow class group sage:K.narrow_class_group().invariants() gp:bnfnarrow(K) magma:NarrowClassGroup(K);

Unit group

comment:Unit group

sage:UK = K.unit_group()

magma:UK, fUK := UnitGroup(K);

oscar:UK, fUK = unit_group(OK)

Rank:	$11$	comment:Unit rank sage:UK.rank() gp:K.fu magma:UnitRank(K); oscar:rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	comment:Generator for roots of unity 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:	$\frac{32984941}{65921084}a^{19}+\frac{192243355}{593289756}a^{18}-\frac{444196291}{131842168}a^{17}-\frac{654830761}{296644878}a^{16}+\frac{539117501}{131842168}a^{15}+\frac{372837491}{131842168}a^{14}-\frac{7757800617}{131842168}a^{13}-\frac{7586576509}{197763252}a^{12}+\frac{11763893885}{131842168}a^{11}+\frac{24070407187}{395526504}a^{10}-\frac{14420711145}{131842168}a^{9}-\frac{15077067671}{197763252}a^{8}+\frac{68284201729}{131842168}a^{7}+\frac{406104906415}{1186579512}a^{6}-\frac{29257245173}{131842168}a^{5}-\frac{101883420907}{593289756}a^{4}+\frac{2656197903}{131842168}a^{3}+\frac{37202480407}{1186579512}a^{2}+\frac{314916415}{65921084}a-\frac{443974471}{593289756}$, $\frac{665556733}{1186579512}a^{19}-\frac{54584101}{197763252}a^{18}-\frac{2242729681}{593289756}a^{17}+\frac{186472513}{98881626}a^{16}+\frac{303481037}{65921084}a^{15}-\frac{321271439}{131842168}a^{14}-\frac{26094896425}{395526504}a^{13}+\frac{4306604937}{131842168}a^{12}+\frac{39729302999}{395526504}a^{11}-\frac{1728556647}{32960542}a^{10}-\frac{24302346925}{197763252}a^{9}+\frac{2138953729}{32960542}a^{8}+\frac{345098993629}{593289756}a^{7}-\frac{115355270497}{395526504}a^{6}-\frac{299505453175}{1186579512}a^{5}+\frac{59674258487}{395526504}a^{4}+\frac{6377289323}{296644878}a^{3}-\frac{4249831769}{197763252}a^{2}-\frac{229591933}{296644878}a+\frac{109824589}{49440813}$, $\frac{25698881}{263684336}a^{19}-\frac{37374878}{148322439}a^{18}-\frac{98831183}{131842168}a^{17}+\frac{2019118499}{1186579512}a^{16}+\frac{187211659}{131842168}a^{15}-\frac{276111125}{131842168}a^{14}-\frac{1607541397}{131842168}a^{13}+\frac{5870649839}{197763252}a^{12}+\frac{467140941}{16480271}a^{11}-\frac{9004471249}{197763252}a^{10}-\frac{2463205929}{65921084}a^{9}+\frac{22207083557}{395526504}a^{8}+\frac{15932942475}{131842168}a^{7}-\frac{311215656413}{1186579512}a^{6}-\frac{18360304109}{131842168}a^{5}+\frac{69479906795}{593289756}a^{4}+\frac{10785772575}{263684336}a^{3}-\frac{8326767049}{593289756}a^{2}-\frac{573005055}{131842168}a-\frac{426569405}{296644878}$, $\frac{119844179}{593289756}a^{19}-\frac{89494709}{1186579512}a^{18}-\frac{218332468}{148322439}a^{17}+\frac{627489385}{1186579512}a^{16}+\frac{158590337}{65921084}a^{15}-\frac{50212793}{65921084}a^{14}-\frac{9762013231}{395526504}a^{13}+\frac{448408267}{49440813}a^{12}+\frac{19460262929}{395526504}a^{11}-\frac{6312242491}{395526504}a^{10}-\frac{12716590357}{197763252}a^{9}+\frac{8192287087}{395526504}a^{8}+\frac{138812174389}{593289756}a^{7}-\frac{49518480665}{593289756}a^{6}-\frac{244375623601}{1186579512}a^{5}+\frac{8346082666}{148322439}a^{4}+\frac{70175095673}{1186579512}a^{3}-\frac{2348585927}{148322439}a^{2}-\frac{3024202931}{593289756}a+\frac{356676763}{148322439}$, $\frac{1269099655}{2373159024}a^{19}+\frac{807479231}{2373159024}a^{18}-\frac{8406601343}{2373159024}a^{17}-\frac{5408123497}{2373159024}a^{16}+\frac{1057091293}{263684336}a^{15}+\frac{714702811}{263684336}a^{14}-\frac{49537454011}{791053008}a^{13}-\frac{31636328921}{791053008}a^{12}+\frac{70189254329}{791053008}a^{11}+\frac{46944334279}{791053008}a^{10}-\frac{87189090509}{791053008}a^{9}-\frac{58225884175}{791053008}a^{8}+\frac{1295947531073}{2373159024}a^{7}+\frac{834304772047}{2373159024}a^{6}-\frac{434033110573}{2373159024}a^{5}-\frac{335065467623}{2373159024}a^{4}+\frac{4164515255}{148322439}a^{3}+\frac{23416699073}{1186579512}a^{2}-\frac{1531279309}{593289756}a-\frac{352732889}{148322439}$, $\frac{1269099655}{2373159024}a^{19}-\frac{807479231}{2373159024}a^{18}-\frac{8406601343}{2373159024}a^{17}+\frac{5408123497}{2373159024}a^{16}+\frac{1057091293}{263684336}a^{15}-\frac{714702811}{263684336}a^{14}-\frac{49537454011}{791053008}a^{13}+\frac{31636328921}{791053008}a^{12}+\frac{70189254329}{791053008}a^{11}-\frac{46944334279}{791053008}a^{10}-\frac{87189090509}{791053008}a^{9}+\frac{58225884175}{791053008}a^{8}+\frac{1295947531073}{2373159024}a^{7}-\frac{834304772047}{2373159024}a^{6}-\frac{434033110573}{2373159024}a^{5}+\frac{335065467623}{2373159024}a^{4}+\frac{4164515255}{148322439}a^{3}-\frac{23416699073}{1186579512}a^{2}-\frac{1531279309}{593289756}a+\frac{352732889}{148322439}$, $\frac{25698881}{263684336}a^{19}+\frac{37374878}{148322439}a^{18}-\frac{98831183}{131842168}a^{17}-\frac{2019118499}{1186579512}a^{16}+\frac{187211659}{131842168}a^{15}+\frac{276111125}{131842168}a^{14}-\frac{1607541397}{131842168}a^{13}-\frac{5870649839}{197763252}a^{12}+\frac{467140941}{16480271}a^{11}+\frac{9004471249}{197763252}a^{10}-\frac{2463205929}{65921084}a^{9}-\frac{22207083557}{395526504}a^{8}+\frac{15932942475}{131842168}a^{7}+\frac{311215656413}{1186579512}a^{6}-\frac{18360304109}{131842168}a^{5}-\frac{69479906795}{593289756}a^{4}+\frac{10785772575}{263684336}a^{3}+\frac{8326767049}{593289756}a^{2}-\frac{573005055}{131842168}a+\frac{426569405}{296644878}$, $\frac{154605061}{1186579512}a^{19}-\frac{28767967}{148322439}a^{18}-\frac{2046336355}{2373159024}a^{17}+\frac{3218680811}{2373159024}a^{16}+\frac{258113305}{263684336}a^{15}-\frac{506372735}{263684336}a^{14}-\frac{12102917171}{791053008}a^{13}+\frac{18350958283}{791053008}a^{12}+\frac{17063539183}{791053008}a^{11}-\frac{32028126269}{791053008}a^{10}-\frac{21795657955}{791053008}a^{9}+\frac{40326209483}{791053008}a^{8}+\frac{317905664029}{2373159024}a^{7}-\frac{501755720675}{2373159024}a^{6}-\frac{105954727637}{2373159024}a^{5}+\frac{326864531029}{2373159024}a^{4}+\frac{31862675407}{2373159024}a^{3}-\frac{64865377331}{2373159024}a^{2}-\frac{418241269}{1186579512}a+\frac{3653039873}{1186579512}$, $\frac{154605061}{1186579512}a^{19}+\frac{28767967}{148322439}a^{18}-\frac{2046336355}{2373159024}a^{17}-\frac{3218680811}{2373159024}a^{16}+\frac{258113305}{263684336}a^{15}+\frac{506372735}{263684336}a^{14}-\frac{12102917171}{791053008}a^{13}-\frac{18350958283}{791053008}a^{12}+\frac{17063539183}{791053008}a^{11}+\frac{32028126269}{791053008}a^{10}-\frac{21795657955}{791053008}a^{9}-\frac{40326209483}{791053008}a^{8}+\frac{317905664029}{2373159024}a^{7}+\frac{501755720675}{2373159024}a^{6}-\frac{105954727637}{2373159024}a^{5}-\frac{326864531029}{2373159024}a^{4}+\frac{31862675407}{2373159024}a^{3}+\frac{64865377331}{2373159024}a^{2}-\frac{418241269}{1186579512}a-\frac{3653039873}{1186579512}$, $\frac{42783989}{1186579512}a^{19}+\frac{416999315}{2373159024}a^{18}-\frac{581468801}{2373159024}a^{17}-\frac{2865159541}{2373159024}a^{16}+\frac{78216161}{263684336}a^{15}+\frac{421697025}{263684336}a^{14}-\frac{3303334711}{791053008}a^{13}-\frac{16507080947}{791053008}a^{12}+\frac{5238764261}{791053008}a^{11}+\frac{27054922255}{791053008}a^{10}-\frac{5360683961}{791053008}a^{9}-\frac{33859731835}{791053008}a^{8}+\frac{85766295293}{2373159024}a^{7}+\frac{445102772821}{2373159024}a^{6}-\frac{38458808425}{2373159024}a^{5}-\frac{245791436789}{2373159024}a^{4}-\frac{24127729135}{2373159024}a^{3}+\frac{22143839015}{1186579512}a^{2}+\frac{428507689}{1186579512}a-\frac{194752826}{148322439}$, $\frac{131974925}{593289756}a^{18}-\frac{1893166313}{1186579512}a^{16}+\frac{327298299}{131842168}a^{14}-\frac{5340446843}{197763252}a^{12}+\frac{2532110956}{49440813}a^{10}-\frac{26446630619}{395526504}a^{8}+\frac{300167242775}{1186579512}a^{6}-\frac{119571922229}{593289756}a^{4}+\frac{35361829315}{593289756}a^{2}-\frac{1858073533}{296644878}$ 32984941/65921084a^19 + 192243355/593289756a^18 - 444196291/131842168a^17 - 654830761/296644878a^16 + 539117501/131842168a^15 + 372837491/131842168a^14 - 7757800617/131842168a^13 - 7586576509/197763252a^12 + 11763893885/131842168a^11 + 24070407187/395526504a^10 - 14420711145/131842168a^9 - 15077067671/197763252a^8 + 68284201729/131842168a^7 + 406104906415/1186579512a^6 - 29257245173/131842168a^5 - 101883420907/593289756a^4 + 2656197903/131842168a^3 + 37202480407/1186579512a^2 + 314916415/65921084a - 443974471/593289756, 665556733/1186579512a^19 - 54584101/197763252a^18 - 2242729681/593289756a^17 + 186472513/98881626a^16 + 303481037/65921084a^15 - 321271439/131842168a^14 - 26094896425/395526504a^13 + 4306604937/131842168a^12 + 39729302999/395526504a^11 - 1728556647/32960542a^10 - 24302346925/197763252a^9 + 2138953729/32960542a^8 + 345098993629/593289756a^7 - 115355270497/395526504a^6 - 299505453175/1186579512a^5 + 59674258487/395526504a^4 + 6377289323/296644878a^3 - 4249831769/197763252a^2 - 229591933/296644878a + 109824589/49440813, 25698881/263684336a^19 - 37374878/148322439a^18 - 98831183/131842168a^17 + 2019118499/1186579512a^16 + 187211659/131842168a^15 - 276111125/131842168a^14 - 1607541397/131842168a^13 + 5870649839/197763252a^12 + 467140941/16480271a^11 - 9004471249/197763252a^10 - 2463205929/65921084a^9 + 22207083557/395526504a^8 + 15932942475/131842168a^7 - 311215656413/1186579512a^6 - 18360304109/131842168a^5 + 69479906795/593289756a^4 + 10785772575/263684336a^3 - 8326767049/593289756a^2 - 573005055/131842168a - 426569405/296644878, 119844179/593289756a^19 - 89494709/1186579512a^18 - 218332468/148322439a^17 + 627489385/1186579512a^16 + 158590337/65921084a^15 - 50212793/65921084a^14 - 9762013231/395526504a^13 + 448408267/49440813a^12 + 19460262929/395526504a^11 - 6312242491/395526504a^10 - 12716590357/197763252a^9 + 8192287087/395526504a^8 + 138812174389/593289756a^7 - 49518480665/593289756a^6 - 244375623601/1186579512a^5 + 8346082666/148322439a^4 + 70175095673/1186579512a^3 - 2348585927/148322439a^2 - 3024202931/593289756a + 356676763/148322439, 1269099655/2373159024a^19 + 807479231/2373159024a^18 - 8406601343/2373159024a^17 - 5408123497/2373159024a^16 + 1057091293/263684336a^15 + 714702811/263684336a^14 - 49537454011/791053008a^13 - 31636328921/791053008a^12 + 70189254329/791053008a^11 + 46944334279/791053008a^10 - 87189090509/791053008a^9 - 58225884175/791053008a^8 + 1295947531073/2373159024a^7 + 834304772047/2373159024a^6 - 434033110573/2373159024a^5 - 335065467623/2373159024a^4 + 4164515255/148322439a^3 + 23416699073/1186579512a^2 - 1531279309/593289756a - 352732889/148322439, 1269099655/2373159024a^19 - 807479231/2373159024a^18 - 8406601343/2373159024a^17 + 5408123497/2373159024a^16 + 1057091293/263684336a^15 - 714702811/263684336a^14 - 49537454011/791053008a^13 + 31636328921/791053008a^12 + 70189254329/791053008a^11 - 46944334279/791053008a^10 - 87189090509/791053008a^9 + 58225884175/791053008a^8 + 1295947531073/2373159024a^7 - 834304772047/2373159024a^6 - 434033110573/2373159024a^5 + 335065467623/2373159024a^4 + 4164515255/148322439a^3 - 23416699073/1186579512a^2 - 1531279309/593289756a + 352732889/148322439, 25698881/263684336a^19 + 37374878/148322439a^18 - 98831183/131842168a^17 - 2019118499/1186579512a^16 + 187211659/131842168a^15 + 276111125/131842168a^14 - 1607541397/131842168a^13 - 5870649839/197763252a^12 + 467140941/16480271a^11 + 9004471249/197763252a^10 - 2463205929/65921084a^9 - 22207083557/395526504a^8 + 15932942475/131842168a^7 + 311215656413/1186579512a^6 - 18360304109/131842168a^5 - 69479906795/593289756a^4 + 10785772575/263684336a^3 + 8326767049/593289756a^2 - 573005055/131842168a + 426569405/296644878, 154605061/1186579512a^19 - 28767967/148322439a^18 - 2046336355/2373159024a^17 + 3218680811/2373159024a^16 + 258113305/263684336a^15 - 506372735/263684336a^14 - 12102917171/791053008a^13 + 18350958283/791053008a^12 + 17063539183/791053008a^11 - 32028126269/791053008a^10 - 21795657955/791053008a^9 + 40326209483/791053008a^8 + 317905664029/2373159024a^7 - 501755720675/2373159024a^6 - 105954727637/2373159024a^5 + 326864531029/2373159024a^4 + 31862675407/2373159024a^3 - 64865377331/2373159024a^2 - 418241269/1186579512a + 3653039873/1186579512, 154605061/1186579512a^19 + 28767967/148322439a^18 - 2046336355/2373159024a^17 - 3218680811/2373159024a^16 + 258113305/263684336a^15 + 506372735/263684336a^14 - 12102917171/791053008a^13 - 18350958283/791053008a^12 + 17063539183/791053008a^11 + 32028126269/791053008a^10 - 21795657955/791053008a^9 - 40326209483/791053008a^8 + 317905664029/2373159024a^7 + 501755720675/2373159024a^6 - 105954727637/2373159024a^5 - 326864531029/2373159024a^4 + 31862675407/2373159024a^3 + 64865377331/2373159024a^2 - 418241269/1186579512a - 3653039873/1186579512, 42783989/1186579512a^19 + 416999315/2373159024a^18 - 581468801/2373159024a^17 - 2865159541/2373159024a^16 + 78216161/263684336a^15 + 421697025/263684336a^14 - 3303334711/791053008a^13 - 16507080947/791053008a^12 + 5238764261/791053008a^11 + 27054922255/791053008a^10 - 5360683961/791053008a^9 - 33859731835/791053008a^8 + 85766295293/2373159024a^7 + 445102772821/2373159024a^6 - 38458808425/2373159024a^5 - 245791436789/2373159024a^4 - 24127729135/2373159024a^3 + 22143839015/1186579512a^2 + 428507689/1186579512a - 194752826/148322439, 131974925/593289756a^18 - 1893166313/1186579512a^16 + 327298299/131842168a^14 - 5340446843/197763252a^12 + 2532110956/49440813a^10 - 26446630619/395526504a^8 + 300167242775/1186579512a^6 - 119571922229/593289756a^4 + 35361829315/593289756*a^2 - 1858073533/296644878 (assuming GRH)	comment:Fundamental units 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:	$ 122691592.972 $ (assuming GRH)	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)
Unit signature rank:	$ 3 $ (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 122691592.972 \cdot 1}{2\cdot\sqrt{4800653894273660658369343520768}}\cr\approx \mathstrut & 1.0881619627 \end{aligned}\] (assuming GRH)

comment:Analytic class number formula

sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^18 + 10*x^16 - 120*x^14 + 210*x^12 - 270*x^10 + 1100*x^8 - 728*x^6 + 193*x^4 - 23*x^2 + 2) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^20 - 7*x^18 + 10*x^16 - 120*x^14 + 210*x^12 - 270*x^10 + 1100*x^8 - 728*x^6 + 193*x^4 - 23*x^2 + 2, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 7*x^18 + 10*x^16 - 120*x^14 + 210*x^12 - 270*x^10 + 1100*x^8 - 728*x^6 + 193*x^4 - 23*x^2 + 2); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 7*x^18 + 10*x^16 - 120*x^14 + 210*x^12 - 270*x^10 + 1100*x^8 - 728*x^6 + 193*x^4 - 23*x^2 + 2); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$C_2^9.S_6:C_2$ (as 20T951):

comment:Galois group

sage:K.galois_group()

gp:polgalois(K.pol)

magma:GaloisGroup(K);

oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)

A non-solvable group of order 737280

The 65 conjugacy class representatives for $C_2^9.S_6:C_2$

Character table for $C_2^9.S_6:C_2$

Intermediate fields

10.2.24207794634752.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

comment:Intermediate fields

sage:K.subfields()[1:-1]

gp:L = nfsubfields(K); L[2..length(L)]

magma:L := Subfields(K); L[2..#L];

oscar:subfields(K)[2:end-1]

Sibling fields

Degree 20 siblings:	data not computed
Degree 40 siblings:	data not computed
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	$16{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$	$20$	${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }$	${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }$	$16{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$	${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{3}$	$16{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$	${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$	$20$	${\href{/padicField/31.5.0.1}{5} }^{4}$	${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$	${\href{/padicField/41.10.0.1}{10} }^{2}$	${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$	${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$	$16{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$	$20$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

comment:Frobenius cycle types

sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	2.1.2.3a1.4	$x^{2} + 4 x + 10$	$2$	$1$	$3$	$C_2$	$$[3]$$
	2.2.1.0a1.1	$x^{2} + x + 1$	$1$	$2$	$0$	$C_2$	$$[\ ]^{2}$$
	2.1.16.58n1.779	$x^{16} + 8 x^{15} + 8 x^{14} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 8 x^{6} + 20 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 14$	$16$	$1$	$58$	16T333	$$[2, 3, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$$
$113$ 113	113.1.2.1a1.1	$x^{2} + 113$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	113.1.2.1a1.1	$x^{2} + 113$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	113.4.1.0a1.1	$x^{4} + 62 x + 3$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
	113.4.1.0a1.1	$x^{4} + 62 x + 3$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
	113.4.2.4a1.2	$x^{8} + 124 x^{5} + 6 x^{4} + 3844 x^{2} + 372 x + 122$	$2$	$4$	$4$	$C_4\times C_2$	$$[\ ]_{2}^{4}$$

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