LMFDB - Number field 24.0.138359014736314946502328332753681.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^23 + 2*x^22 + 10*x^21 - 20*x^20 + 13*x^19 + 57*x^18 - 98*x^17 + 19*x^16 + 228*x^15 - 267*x^14 - 159*x^13 + 711*x^12 - 318*x^11 - 1068*x^10 + 1824*x^9 + 304*x^8 - 3136*x^7 + 3648*x^6 + 1664*x^5 - 5120*x^4 + 5120*x^3 + 2048*x^2 - 4096*x + 4096)

gp: K = bnfinit(y^24 - 2*y^23 + 2*y^22 + 10*y^21 - 20*y^20 + 13*y^19 + 57*y^18 - 98*y^17 + 19*y^16 + 228*y^15 - 267*y^14 - 159*y^13 + 711*y^12 - 318*y^11 - 1068*y^10 + 1824*y^9 + 304*y^8 - 3136*y^7 + 3648*y^6 + 1664*y^5 - 5120*y^4 + 5120*y^3 + 2048*y^2 - 4096*y + 4096, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 2*x^23 + 2*x^22 + 10*x^21 - 20*x^20 + 13*x^19 + 57*x^18 - 98*x^17 + 19*x^16 + 228*x^15 - 267*x^14 - 159*x^13 + 711*x^12 - 318*x^11 - 1068*x^10 + 1824*x^9 + 304*x^8 - 3136*x^7 + 3648*x^6 + 1664*x^5 - 5120*x^4 + 5120*x^3 + 2048*x^2 - 4096*x + 4096);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 2*x^23 + 2*x^22 + 10*x^21 - 20*x^20 + 13*x^19 + 57*x^18 - 98*x^17 + 19*x^16 + 228*x^15 - 267*x^14 - 159*x^13 + 711*x^12 - 318*x^11 - 1068*x^10 + 1824*x^9 + 304*x^8 - 3136*x^7 + 3648*x^6 + 1664*x^5 - 5120*x^4 + 5120*x^3 + 2048*x^2 - 4096*x + 4096)

$ x^{24} - 2 x^{23} + 2 x^{22} + 10 x^{21} - 20 x^{20} + 13 x^{19} + 57 x^{18} - 98 x^{17} + 19 x^{16} + 228 x^{15} - 267 x^{14} - 159 x^{13} + 711 x^{12} - 318 x^{11} - 1068 x^{10} + 1824 x^{9} + \cdots + 4096 $ x^24 - 2*x^23 + 2*x^22 + 10*x^21 - 20*x^20 + 13*x^19 + 57*x^18 - 98*x^17 + 19*x^16 + 228*x^15 - 267*x^14 - 159*x^13 + 711*x^12 - 318*x^11 - 1068*x^10 + 1824*x^9 + 304*x^8 - 3136*x^7 + 3648*x^6 + 1664*x^5 - 5120*x^4 + 5120*x^3 + 2048*x^2 - 4096*x + 4096

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$24$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 12]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$138359014736314946502328332753681$ 138359014736314946502328332753681 $\medspace = 3^{12}\cdot 7^{20}\cdot 239^{4}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$21.84$	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))
Ramified primes:	$3$, $7$, $239$ 3, 7, 239	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$8$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:	unavailable$^{2048}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{10}+\frac{1}{8}a^{9}+\frac{1}{4}a^{8}+\frac{1}{8}a^{7}+\frac{1}{4}a^{6}+\frac{1}{8}a^{5}-\frac{1}{8}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{16}-\frac{1}{8}a^{14}-\frac{1}{8}a^{13}-\frac{3}{16}a^{11}+\frac{3}{16}a^{10}-\frac{1}{16}a^{8}+\frac{1}{8}a^{7}+\frac{5}{16}a^{6}-\frac{5}{16}a^{5}-\frac{7}{16}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{32}a^{17}-\frac{1}{16}a^{15}-\frac{1}{16}a^{14}-\frac{3}{32}a^{12}-\frac{5}{32}a^{11}+\frac{7}{32}a^{9}-\frac{7}{16}a^{8}-\frac{11}{32}a^{7}+\frac{3}{32}a^{6}-\frac{7}{32}a^{5}-\frac{1}{4}a^{3}$, $\frac{1}{64}a^{18}-\frac{1}{32}a^{16}-\frac{1}{32}a^{15}+\frac{5}{64}a^{13}+\frac{3}{64}a^{12}-\frac{1}{64}a^{10}+\frac{5}{32}a^{9}-\frac{19}{64}a^{8}-\frac{21}{64}a^{7}+\frac{25}{64}a^{6}-\frac{3}{8}a^{4}-\frac{1}{8}a^{3}$, $\frac{1}{128}a^{19}-\frac{1}{64}a^{17}-\frac{1}{64}a^{16}+\frac{5}{128}a^{14}-\frac{13}{128}a^{13}-\frac{1}{8}a^{12}+\frac{31}{128}a^{11}+\frac{13}{64}a^{10}+\frac{29}{128}a^{9}+\frac{59}{128}a^{8}+\frac{9}{128}a^{7}+\frac{1}{4}a^{6}-\frac{3}{16}a^{5}-\frac{1}{16}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{512}a^{20}-\frac{1}{256}a^{19}-\frac{1}{256}a^{18}+\frac{1}{256}a^{17}+\frac{1}{128}a^{16}+\frac{5}{512}a^{15}+\frac{41}{512}a^{14}+\frac{21}{256}a^{13}-\frac{33}{512}a^{12}+\frac{23}{128}a^{11}-\frac{119}{512}a^{10}+\frac{33}{512}a^{9}-\frac{141}{512}a^{8}-\frac{41}{256}a^{7}-\frac{3}{64}a^{6}+\frac{29}{64}a^{5}-\frac{1}{8}a^{4}+\frac{1}{8}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{1024}a^{21}+\frac{1}{512}a^{19}-\frac{1}{512}a^{18}-\frac{1}{128}a^{17}-\frac{3}{1024}a^{16}+\frac{51}{1024}a^{15}+\frac{9}{256}a^{14}+\frac{75}{1024}a^{13}-\frac{51}{512}a^{12}+\frac{185}{1024}a^{11}-\frac{125}{1024}a^{10}-\frac{227}{1024}a^{9}+\frac{123}{256}a^{8}-\frac{29}{256}a^{7}-\frac{41}{128}a^{6}+\frac{29}{64}a^{5}+\frac{7}{16}a^{4}-\frac{5}{16}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{176128}a^{22}+\frac{19}{44032}a^{21}+\frac{83}{88064}a^{20}+\frac{39}{88064}a^{19}-\frac{165}{22016}a^{18}+\frac{613}{176128}a^{17}-\frac{2465}{176128}a^{16}+\frac{127}{44032}a^{15}+\frac{4927}{176128}a^{14}-\frac{10685}{88064}a^{13}-\frac{11763}{176128}a^{12}-\frac{41249}{176128}a^{11}+\frac{12293}{176128}a^{10}+\frac{8171}{44032}a^{9}-\frac{319}{22016}a^{8}+\frac{343}{1376}a^{7}-\frac{5403}{11008}a^{6}+\frac{2301}{5504}a^{5}+\frac{155}{2752}a^{4}-\frac{575}{1376}a^{3}-\frac{3}{172}a^{2}+\frac{109}{344}a+\frac{45}{172}$, $\frac{1}{44736512}a^{23}+\frac{7}{22368256}a^{22}-\frac{6745}{22368256}a^{21}-\frac{17491}{22368256}a^{20}-\frac{10139}{11184128}a^{19}+\frac{28789}{44736512}a^{18}-\frac{656919}{44736512}a^{17}+\frac{7599}{520192}a^{16}-\frac{2742105}{44736512}a^{15}+\frac{789297}{11184128}a^{14}-\frac{39205}{1040384}a^{13}-\frac{2106599}{44736512}a^{12}+\frac{67475}{44736512}a^{11}-\frac{2905869}{22368256}a^{10}-\frac{305561}{2796032}a^{9}-\frac{1375407}{2796032}a^{8}+\frac{714569}{2796032}a^{7}-\frac{303565}{699008}a^{6}+\frac{15273}{174752}a^{5}+\frac{741}{5461}a^{4}-\frac{42903}{174752}a^{3}+\frac{33849}{87376}a^{2}-\frac{7859}{21844}a+\frac{2905}{21844}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, 1/2*a^9 - 1/2*a^7 - 1/2*a^4 - 1/2*a^2 - 1/2*a, 1/2*a^10 - 1/2*a^8 - 1/2*a^5 - 1/2*a^3 - 1/2*a^2, 1/2*a^11 - 1/2*a^7 - 1/2*a^6 - 1/2*a^3 - 1/2*a^2 - 1/2*a, 1/4*a^12 - 1/4*a^11 - 1/4*a^9 - 1/4*a^8 + 1/4*a^7 - 1/4*a^6 + 1/4*a^2 - 1/2*a, 1/4*a^13 - 1/4*a^11 - 1/4*a^10 - 1/2*a^7 - 1/4*a^6 - 1/2*a^4 + 1/4*a^3 + 1/4*a^2, 1/4*a^14 - 1/4*a^9 + 1/4*a^8 - 1/2*a^7 + 1/4*a^6 - 1/2*a^5 + 1/4*a^4 - 1/4*a^3 - 1/4*a^2, 1/8*a^15 - 1/8*a^10 + 1/8*a^9 + 1/4*a^8 + 1/8*a^7 + 1/4*a^6 + 1/8*a^5 - 1/8*a^4 + 3/8*a^3 - 1/2*a^2, 1/16*a^16 - 1/8*a^14 - 1/8*a^13 - 3/16*a^11 + 3/16*a^10 - 1/16*a^8 + 1/8*a^7 + 5/16*a^6 - 5/16*a^5 - 7/16*a^4 - 1/2*a^3, 1/32*a^17 - 1/16*a^15 - 1/16*a^14 - 3/32*a^12 - 5/32*a^11 + 7/32*a^9 - 7/16*a^8 - 11/32*a^7 + 3/32*a^6 - 7/32*a^5 - 1/4*a^3, 1/64*a^18 - 1/32*a^16 - 1/32*a^15 + 5/64*a^13 + 3/64*a^12 - 1/64*a^10 + 5/32*a^9 - 19/64*a^8 - 21/64*a^7 + 25/64*a^6 - 3/8*a^4 - 1/8*a^3, 1/128*a^19 - 1/64*a^17 - 1/64*a^16 + 5/128*a^14 - 13/128*a^13 - 1/8*a^12 + 31/128*a^11 + 13/64*a^10 + 29/128*a^9 + 59/128*a^8 + 9/128*a^7 + 1/4*a^6 - 3/16*a^5 - 1/16*a^4 + 3/8*a^3 - 1/2*a, 1/512*a^20 - 1/256*a^19 - 1/256*a^18 + 1/256*a^17 + 1/128*a^16 + 5/512*a^15 + 41/512*a^14 + 21/256*a^13 - 33/512*a^12 + 23/128*a^11 - 119/512*a^10 + 33/512*a^9 - 141/512*a^8 - 41/256*a^7 - 3/64*a^6 + 29/64*a^5 - 1/8*a^4 + 1/8*a^2 - 1/2*a - 1/2, 1/1024*a^21 + 1/512*a^19 - 1/512*a^18 - 1/128*a^17 - 3/1024*a^16 + 51/1024*a^15 + 9/256*a^14 + 75/1024*a^13 - 51/512*a^12 + 185/1024*a^11 - 125/1024*a^10 - 227/1024*a^9 + 123/256*a^8 - 29/256*a^7 - 41/128*a^6 + 29/64*a^5 + 7/16*a^4 - 5/16*a^3 + 1/8*a^2 + 1/4*a - 1/2, 1/176128*a^22 + 19/44032*a^21 + 83/88064*a^20 + 39/88064*a^19 - 165/22016*a^18 + 613/176128*a^17 - 2465/176128*a^16 + 127/44032*a^15 + 4927/176128*a^14 - 10685/88064*a^13 - 11763/176128*a^12 - 41249/176128*a^11 + 12293/176128*a^10 + 8171/44032*a^9 - 319/22016*a^8 + 343/1376*a^7 - 5403/11008*a^6 + 2301/5504*a^5 + 155/2752*a^4 - 575/1376*a^3 - 3/172*a^2 + 109/344*a + 45/172, 1/44736512*a^23 + 7/22368256*a^22 - 6745/22368256*a^21 - 17491/22368256*a^20 - 10139/11184128*a^19 + 28789/44736512*a^18 - 656919/44736512*a^17 + 7599/520192*a^16 - 2742105/44736512*a^15 + 789297/11184128*a^14 - 39205/1040384*a^13 - 2106599/44736512*a^12 + 67475/44736512*a^11 - 2905869/22368256*a^10 - 305561/2796032*a^9 - 1375407/2796032*a^8 + 714569/2796032*a^7 - 303565/699008*a^6 + 15273/174752*a^5 + 741/5461*a^4 - 42903/174752*a^3 + 33849/87376*a^2 - 7859/21844*a + 2905/21844

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

$C_{3}$, which has order $3$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

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

oscar: UK, fUK = unit_group(OK)

Rank:	$11$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ \frac{233629}{22368256} a^{23} + \frac{356283}{22368256} a^{22} - \frac{244523}{11184128} a^{21} + \frac{59507}{699008} a^{20} + \frac{1973597}{11184128} a^{19} - \frac{5059799}{22368256} a^{18} + \frac{2173295}{11184128} a^{17} + \frac{22093205}{22368256} a^{16} - \frac{502931}{520192} a^{15} - \frac{13893057}{22368256} a^{14} + \frac{77191467}{22368256} a^{13} - \frac{16518885}{11184128} a^{12} - \frac{59569811}{11184128} a^{11} + \frac{181994879}{22368256} a^{10} + \frac{23259973}{5592064} a^{9} - \frac{43889855}{2796032} a^{8} + \frac{18372467}{1398016} a^{7} + \frac{35452205}{1398016} a^{6} - \frac{16496667}{699008} a^{5} + \frac{4517191}{349504} a^{4} + \frac{9277671}{174752} a^{3} - \frac{653149}{43688} a^{2} + \frac{263005}{43688} a + \frac{1000995}{21844} $ (233629)/(22368256)a^(23) + (356283)/(22368256)a^(22) - (244523)/(11184128)a^(21) + (59507)/(699008)a^(20) + (1973597)/(11184128)a^(19) - (5059799)/(22368256)a^(18) + (2173295)/(11184128)a^(17) + (22093205)/(22368256)a^(16) - (502931)/(520192)a^(15) - (13893057)/(22368256)a^(14) + (77191467)/(22368256)a^(13) - (16518885)/(11184128)a^(12) - (59569811)/(11184128)a^(11) + (181994879)/(22368256)a^(10) + (23259973)/(5592064)a^(9) - (43889855)/(2796032)a^(8) + (18372467)/(1398016)a^(7) + (35452205)/(1398016)a^(6) - (16496667)/(699008)a^(5) + (4517191)/(349504)a^(4) + (9277671)/(174752)a^(3) - (653149)/(43688)a^(2) + (263005)/(43688)*a + (1000995)/(21844) (order $42$)	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:	$\frac{489923}{44736512}a^{23}-\frac{831389}{22368256}a^{22}+\frac{151605}{22368256}a^{21}+\frac{2917971}{22368256}a^{20}-\frac{3919615}{11184128}a^{19}-\frac{1031425}{44736512}a^{18}+\frac{39186535}{44736512}a^{17}-\frac{33189959}{22368256}a^{16}-\frac{32898683}{44736512}a^{15}+\frac{5239301}{1398016}a^{14}-\frac{130082829}{44736512}a^{13}-\frac{236457753}{44736512}a^{12}+\frac{471863437}{44736512}a^{11}+\frac{18022431}{22368256}a^{10}-\frac{15041351}{699008}a^{9}+\frac{26654251}{1398016}a^{8}+\frac{50189805}{2796032}a^{7}-\frac{19068869}{349504}a^{6}+\frac{3359763}{174752}a^{5}+\frac{7241099}{174752}a^{4}-\frac{14665679}{174752}a^{3}+\frac{535345}{87376}a^{2}+\frac{208087}{5461}a-\frac{1441905}{21844}$, $\frac{186711}{11184128}a^{23}-\frac{170287}{5592064}a^{22}-\frac{20233}{5592064}a^{21}+\frac{1024863}{5592064}a^{20}-\frac{760543}{2796032}a^{19}-\frac{1553669}{11184128}a^{18}+\frac{11506911}{11184128}a^{17}-\frac{5702975}{5592064}a^{16}-\frac{14403443}{11184128}a^{15}+\frac{10046929}{2796032}a^{14}-\frac{12853773}{11184128}a^{13}-\frac{71296945}{11184128}a^{12}+\frac{90670289}{11184128}a^{11}+\frac{30163259}{5592064}a^{10}-\frac{56807143}{2796032}a^{9}+\frac{15618205}{1398016}a^{8}+\frac{2289589}{87376}a^{7}-\frac{7456397}{174752}a^{6}+\frac{1182983}{174752}a^{5}+\frac{274461}{5461}a^{4}-\frac{308375}{5461}a^{3}-\frac{25141}{10922}a^{2}+\frac{228026}{5461}a-\frac{215093}{5461}$, $\frac{134005}{44736512}a^{23}-\frac{74595}{2796032}a^{22}+\frac{261975}{22368256}a^{21}+\frac{1084391}{22368256}a^{20}-\frac{716937}{2796032}a^{19}+\frac{2550249}{44736512}a^{18}+\frac{20008239}{44736512}a^{17}-\frac{6398457}{5592064}a^{16}-\frac{6187765}{44736512}a^{15}+\frac{52586189}{22368256}a^{14}-\frac{118750039}{44736512}a^{13}-\frac{118308993}{44736512}a^{12}+\frac{339682453}{44736512}a^{11}-\frac{5144979}{2796032}a^{10}-\frac{76084771}{5592064}a^{9}+\frac{42807823}{2796032}a^{8}+\frac{18256447}{2796032}a^{7}-\frac{53567841}{1398016}a^{6}+\frac{12073355}{699008}a^{5}+\frac{6978059}{349504}a^{4}-\frac{5437783}{87376}a^{3}+\frac{632403}{87376}a^{2}+\frac{879567}{43688}a-\frac{268431}{5461}$, $\frac{730331}{22368256}a^{23}-\frac{898407}{22368256}a^{22}-\frac{145533}{11184128}a^{21}+\frac{486177}{1398016}a^{20}-\frac{3940329}{11184128}a^{19}-\frac{7376225}{22368256}a^{18}+\frac{20614079}{11184128}a^{17}-\frac{26629977}{22368256}a^{16}-\frac{58612991}{22368256}a^{15}+\frac{130991685}{22368256}a^{14}-\frac{5638907}{22368256}a^{13}-\frac{129615177}{11184128}a^{12}+\frac{2928071}{260096}a^{11}+\frac{294326813}{22368256}a^{10}-\frac{182486611}{5592064}a^{9}+\frac{15684997}{1398016}a^{8}+\frac{18296691}{349504}a^{7}-\frac{83364361}{1398016}a^{6}-\frac{395811}{699008}a^{5}+\frac{34182889}{349504}a^{4}-\frac{11789367}{174752}a^{3}-\frac{251985}{21844}a^{2}+\frac{3614327}{43688}a-\frac{905689}{21844}$, $\frac{1244123}{44736512}a^{23}-\frac{40555}{1398016}a^{22}-\frac{381307}{22368256}a^{21}+\frac{6532441}{22368256}a^{20}-\frac{333119}{1398016}a^{19}-\frac{14609161}{44736512}a^{18}+\frac{66264497}{44736512}a^{17}-\frac{1889719}{2796032}a^{16}-\frac{104380883}{44736512}a^{15}+\frac{97469083}{22368256}a^{14}+\frac{36027087}{44736512}a^{13}-\frac{422791535}{44736512}a^{12}+\frac{320543971}{44736512}a^{11}+\frac{70623125}{5592064}a^{10}-\frac{67532219}{2796032}a^{9}+\frac{10725751}{2796032}a^{8}+\frac{122469447}{2796032}a^{7}-\frac{54431957}{1398016}a^{6}-\frac{5834733}{699008}a^{5}+\frac{26806009}{349504}a^{4}-\frac{201999}{5461}a^{3}-\frac{1281385}{87376}a^{2}+\frac{2682535}{43688}a-\frac{202305}{10922}$, $\frac{3273}{699008}a^{23}+\frac{109401}{5592064}a^{22}-\frac{81515}{5592064}a^{21}+\frac{4001}{174752}a^{20}+\frac{6535}{32512}a^{19}-\frac{370899}{2796032}a^{18}-\frac{583183}{5592064}a^{17}+\frac{1389861}{1398016}a^{16}-\frac{2299375}{5592064}a^{15}-\frac{7608131}{5592064}a^{14}+\frac{15899645}{5592064}a^{13}+\frac{2608877}{5592064}a^{12}-\frac{129757}{21844}a^{11}+\frac{13154929}{2796032}a^{10}+\frac{45791091}{5592064}a^{9}-\frac{39453419}{2796032}a^{8}+\frac{1193115}{349504}a^{7}+\frac{20640611}{699008}a^{6}-\frac{1619775}{87376}a^{5}-\frac{181437}{87376}a^{4}+\frac{4627745}{87376}a^{3}-\frac{56667}{5461}a^{2}-\frac{105437}{21844}a+\frac{230309}{5461}$, $\frac{173969}{22368256}a^{23}+\frac{643743}{11184128}a^{22}-\frac{500975}{11184128}a^{21}+\frac{341657}{11184128}a^{20}+\frac{3277683}{5592064}a^{19}-\frac{8326931}{22368256}a^{18}-\frac{9498279}{22368256}a^{17}+\frac{32149455}{11184128}a^{16}-\frac{22707989}{22368256}a^{15}-\frac{23512993}{5592064}a^{14}+\frac{181974085}{22368256}a^{13}+\frac{45361513}{22368256}a^{12}-\frac{396139833}{22368256}a^{11}+\frac{142532357}{11184128}a^{10}+\frac{140011955}{5592064}a^{9}-\frac{117412515}{2796032}a^{8}+\frac{8872591}{1398016}a^{7}+\frac{60839677}{699008}a^{6}-\frac{9660371}{174752}a^{5}-\frac{1252459}{87376}a^{4}+\frac{6777939}{43688}a^{3}-\frac{1320937}{43688}a^{2}-\frac{518917}{21844}a+\frac{1382985}{10922}$, $\frac{349637}{44736512}a^{23}+\frac{99737}{22368256}a^{22}-\frac{93937}{22368256}a^{21}+\frac{1624413}{22368256}a^{20}+\frac{560831}{11184128}a^{19}-\frac{3890631}{44736512}a^{18}+\frac{12620937}{44736512}a^{17}+\frac{6830103}{22368256}a^{16}-\frac{24684389}{44736512}a^{15}+\frac{2116431}{5592064}a^{14}+\frac{58754797}{44736512}a^{13}-\frac{76200935}{44736512}a^{12}-\frac{40602053}{44736512}a^{11}+\frac{99709713}{22368256}a^{10}-\frac{9122525}{5592064}a^{9}-\frac{6482325}{1398016}a^{8}+\frac{31550885}{2796032}a^{7}+\frac{85029}{16256}a^{6}-\frac{2730739}{349504}a^{5}+\frac{3285903}{174752}a^{4}+\frac{78439}{4064}a^{3}-\frac{375391}{87376}a^{2}+\frac{358611}{21844}a+\frac{463835}{21844}$, $\frac{83821}{11184128}a^{23}-\frac{15995}{5592064}a^{22}-\frac{8479}{699008}a^{21}+\frac{439709}{5592064}a^{20}-\frac{9869}{699008}a^{19}-\frac{1675043}{11184128}a^{18}+\frac{4219881}{11184128}a^{17}+\frac{27335}{699008}a^{16}-\frac{8941991}{11184128}a^{15}+\frac{1310763}{1398016}a^{14}+\frac{8849851}{11184128}a^{13}-\frac{28555619}{11184128}a^{12}+\frac{9360573}{11184128}a^{11}+\frac{1541543}{349504}a^{10}-\frac{29120711}{5592064}a^{9}-\frac{2651881}{1398016}a^{8}+\frac{18562581}{1398016}a^{7}-\frac{4716031}{699008}a^{6}-\frac{2417435}{349504}a^{5}+\frac{2038403}{87376}a^{4}-\frac{563199}{87376}a^{3}-\frac{306317}{43688}a^{2}+\frac{437321}{21844}a-\frac{58325}{10922}$, $\frac{1162609}{44736512}a^{23}+\frac{296539}{5592064}a^{22}-\frac{1655905}{22368256}a^{21}+\frac{4422147}{22368256}a^{20}+\frac{1671683}{2796032}a^{19}-\frac{31383867}{44736512}a^{18}+\frac{11822923}{44736512}a^{17}+\frac{18263247}{5592064}a^{16}-\frac{120945945}{44736512}a^{15}-\frac{66019099}{22368256}a^{14}+\frac{484184093}{44736512}a^{13}-\frac{122101493}{44736512}a^{12}-\frac{829338415}{44736512}a^{11}+\frac{64890479}{2796032}a^{10}+\frac{51238963}{2796032}a^{9}-\frac{144115637}{2796032}a^{8}+\frac{87974793}{2796032}a^{7}+\frac{121566545}{1398016}a^{6}-\frac{53327807}{699008}a^{5}+\frac{7745131}{349504}a^{4}+\frac{1847455}{10922}a^{3}-\frac{4271911}{87376}a^{2}+\frac{124593}{43688}a+\frac{1527021}{10922}$, $\frac{129293}{22368256}a^{23}-\frac{266221}{22368256}a^{22}+\frac{29747}{11184128}a^{21}+\frac{41965}{699008}a^{20}-\frac{1267039}{11184128}a^{19}-\frac{458623}{22368256}a^{18}+\frac{3828851}{11184128}a^{17}-\frac{10454455}{22368256}a^{16}-\frac{7813757}{22368256}a^{15}+\frac{29194663}{22368256}a^{14}-\frac{17945473}{22368256}a^{13}-\frac{23276989}{11184128}a^{12}+\frac{38553015}{11184128}a^{11}+\frac{22254843}{22368256}a^{10}-\frac{10375893}{1398016}a^{9}+\frac{17019693}{2796032}a^{8}+\frac{2728109}{349504}a^{7}-\frac{24416857}{1398016}a^{6}+\frac{3998411}{699008}a^{5}+\frac{5506827}{349504}a^{4}-\frac{4624383}{174752}a^{3}+\frac{80}{127}a^{2}+\frac{578143}{43688}a-\frac{456535}{21844}$ 489923/44736512a^23 - 831389/22368256a^22 + 151605/22368256a^21 + 2917971/22368256a^20 - 3919615/11184128a^19 - 1031425/44736512a^18 + 39186535/44736512a^17 - 33189959/22368256a^16 - 32898683/44736512a^15 + 5239301/1398016a^14 - 130082829/44736512a^13 - 236457753/44736512a^12 + 471863437/44736512a^11 + 18022431/22368256a^10 - 15041351/699008a^9 + 26654251/1398016a^8 + 50189805/2796032a^7 - 19068869/349504a^6 + 3359763/174752a^5 + 7241099/174752a^4 - 14665679/174752a^3 + 535345/87376a^2 + 208087/5461a - 1441905/21844, 186711/11184128a^23 - 170287/5592064a^22 - 20233/5592064a^21 + 1024863/5592064a^20 - 760543/2796032a^19 - 1553669/11184128a^18 + 11506911/11184128a^17 - 5702975/5592064a^16 - 14403443/11184128a^15 + 10046929/2796032a^14 - 12853773/11184128a^13 - 71296945/11184128a^12 + 90670289/11184128a^11 + 30163259/5592064a^10 - 56807143/2796032a^9 + 15618205/1398016a^8 + 2289589/87376a^7 - 7456397/174752a^6 + 1182983/174752a^5 + 274461/5461a^4 - 308375/5461a^3 - 25141/10922a^2 + 228026/5461a - 215093/5461, 134005/44736512a^23 - 74595/2796032a^22 + 261975/22368256a^21 + 1084391/22368256a^20 - 716937/2796032a^19 + 2550249/44736512a^18 + 20008239/44736512a^17 - 6398457/5592064a^16 - 6187765/44736512a^15 + 52586189/22368256a^14 - 118750039/44736512a^13 - 118308993/44736512a^12 + 339682453/44736512a^11 - 5144979/2796032a^10 - 76084771/5592064a^9 + 42807823/2796032a^8 + 18256447/2796032a^7 - 53567841/1398016a^6 + 12073355/699008a^5 + 6978059/349504a^4 - 5437783/87376a^3 + 632403/87376a^2 + 879567/43688a - 268431/5461, 730331/22368256a^23 - 898407/22368256a^22 - 145533/11184128a^21 + 486177/1398016a^20 - 3940329/11184128a^19 - 7376225/22368256a^18 + 20614079/11184128a^17 - 26629977/22368256a^16 - 58612991/22368256a^15 + 130991685/22368256a^14 - 5638907/22368256a^13 - 129615177/11184128a^12 + 2928071/260096a^11 + 294326813/22368256a^10 - 182486611/5592064a^9 + 15684997/1398016a^8 + 18296691/349504a^7 - 83364361/1398016a^6 - 395811/699008a^5 + 34182889/349504a^4 - 11789367/174752a^3 - 251985/21844a^2 + 3614327/43688a - 905689/21844, 1244123/44736512a^23 - 40555/1398016a^22 - 381307/22368256a^21 + 6532441/22368256a^20 - 333119/1398016a^19 - 14609161/44736512a^18 + 66264497/44736512a^17 - 1889719/2796032a^16 - 104380883/44736512a^15 + 97469083/22368256a^14 + 36027087/44736512a^13 - 422791535/44736512a^12 + 320543971/44736512a^11 + 70623125/5592064a^10 - 67532219/2796032a^9 + 10725751/2796032a^8 + 122469447/2796032a^7 - 54431957/1398016a^6 - 5834733/699008a^5 + 26806009/349504a^4 - 201999/5461a^3 - 1281385/87376a^2 + 2682535/43688a - 202305/10922, 3273/699008a^23 + 109401/5592064a^22 - 81515/5592064a^21 + 4001/174752a^20 + 6535/32512a^19 - 370899/2796032a^18 - 583183/5592064a^17 + 1389861/1398016a^16 - 2299375/5592064a^15 - 7608131/5592064a^14 + 15899645/5592064a^13 + 2608877/5592064a^12 - 129757/21844a^11 + 13154929/2796032a^10 + 45791091/5592064a^9 - 39453419/2796032a^8 + 1193115/349504a^7 + 20640611/699008a^6 - 1619775/87376a^5 - 181437/87376a^4 + 4627745/87376a^3 - 56667/5461a^2 - 105437/21844a + 230309/5461, 173969/22368256a^23 + 643743/11184128a^22 - 500975/11184128a^21 + 341657/11184128a^20 + 3277683/5592064a^19 - 8326931/22368256a^18 - 9498279/22368256a^17 + 32149455/11184128a^16 - 22707989/22368256a^15 - 23512993/5592064a^14 + 181974085/22368256a^13 + 45361513/22368256a^12 - 396139833/22368256a^11 + 142532357/11184128a^10 + 140011955/5592064a^9 - 117412515/2796032a^8 + 8872591/1398016a^7 + 60839677/699008a^6 - 9660371/174752a^5 - 1252459/87376a^4 + 6777939/43688a^3 - 1320937/43688a^2 - 518917/21844a + 1382985/10922, 349637/44736512a^23 + 99737/22368256a^22 - 93937/22368256a^21 + 1624413/22368256a^20 + 560831/11184128a^19 - 3890631/44736512a^18 + 12620937/44736512a^17 + 6830103/22368256a^16 - 24684389/44736512a^15 + 2116431/5592064a^14 + 58754797/44736512a^13 - 76200935/44736512a^12 - 40602053/44736512a^11 + 99709713/22368256a^10 - 9122525/5592064a^9 - 6482325/1398016a^8 + 31550885/2796032a^7 + 85029/16256a^6 - 2730739/349504a^5 + 3285903/174752a^4 + 78439/4064a^3 - 375391/87376a^2 + 358611/21844a + 463835/21844, 83821/11184128a^23 - 15995/5592064a^22 - 8479/699008a^21 + 439709/5592064a^20 - 9869/699008a^19 - 1675043/11184128a^18 + 4219881/11184128a^17 + 27335/699008a^16 - 8941991/11184128a^15 + 1310763/1398016a^14 + 8849851/11184128a^13 - 28555619/11184128a^12 + 9360573/11184128a^11 + 1541543/349504a^10 - 29120711/5592064a^9 - 2651881/1398016a^8 + 18562581/1398016a^7 - 4716031/699008a^6 - 2417435/349504a^5 + 2038403/87376a^4 - 563199/87376a^3 - 306317/43688a^2 + 437321/21844a - 58325/10922, 1162609/44736512a^23 + 296539/5592064a^22 - 1655905/22368256a^21 + 4422147/22368256a^20 + 1671683/2796032a^19 - 31383867/44736512a^18 + 11822923/44736512a^17 + 18263247/5592064a^16 - 120945945/44736512a^15 - 66019099/22368256a^14 + 484184093/44736512a^13 - 122101493/44736512a^12 - 829338415/44736512a^11 + 64890479/2796032a^10 + 51238963/2796032a^9 - 144115637/2796032a^8 + 87974793/2796032a^7 + 121566545/1398016a^6 - 53327807/699008a^5 + 7745131/349504a^4 + 1847455/10922a^3 - 4271911/87376a^2 + 124593/43688a + 1527021/10922, 129293/22368256a^23 - 266221/22368256a^22 + 29747/11184128a^21 + 41965/699008a^20 - 1267039/11184128a^19 - 458623/22368256a^18 + 3828851/11184128a^17 - 10454455/22368256a^16 - 7813757/22368256a^15 + 29194663/22368256a^14 - 17945473/22368256a^13 - 23276989/11184128a^12 + 38553015/11184128a^11 + 22254843/22368256a^10 - 10375893/1398016a^9 + 17019693/2796032a^8 + 2728109/349504a^7 - 24416857/1398016a^6 + 3998411/699008a^5 + 5506827/349504a^4 - 4624383/174752a^3 + 80/127a^2 + 578143/43688*a - 456535/21844 (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:	$ 9137650.497592166 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 9137650.497592166 \cdot 3}{42\cdot\sqrt{138359014736314946502328332753681}}\cr\approx \mathstrut & 0.210068627107942 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^23 + 2*x^22 + 10*x^21 - 20*x^20 + 13*x^19 + 57*x^18 - 98*x^17 + 19*x^16 + 228*x^15 - 267*x^14 - 159*x^13 + 711*x^12 - 318*x^11 - 1068*x^10 + 1824*x^9 + 304*x^8 - 3136*x^7 + 3648*x^6 + 1664*x^5 - 5120*x^4 + 5120*x^3 + 2048*x^2 - 4096*x + 4096)

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))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^24 - 2*x^23 + 2*x^22 + 10*x^21 - 20*x^20 + 13*x^19 + 57*x^18 - 98*x^17 + 19*x^16 + 228*x^15 - 267*x^14 - 159*x^13 + 711*x^12 - 318*x^11 - 1068*x^10 + 1824*x^9 + 304*x^8 - 3136*x^7 + 3648*x^6 + 1664*x^5 - 5120*x^4 + 5120*x^3 + 2048*x^2 - 4096*x + 4096, 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)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^24 - 2*x^23 + 2*x^22 + 10*x^21 - 20*x^20 + 13*x^19 + 57*x^18 - 98*x^17 + 19*x^16 + 228*x^15 - 267*x^14 - 159*x^13 + 711*x^12 - 318*x^11 - 1068*x^10 + 1824*x^9 + 304*x^8 - 3136*x^7 + 3648*x^6 + 1664*x^5 - 5120*x^4 + 5120*x^3 + 2048*x^2 - 4096*x + 4096);

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)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 2*x^23 + 2*x^22 + 10*x^21 - 20*x^20 + 13*x^19 + 57*x^18 - 98*x^17 + 19*x^16 + 228*x^15 - 267*x^14 - 159*x^13 + 711*x^12 - 318*x^11 - 1068*x^10 + 1824*x^9 + 304*x^8 - 3136*x^7 + 3648*x^6 + 1664*x^5 - 5120*x^4 + 5120*x^3 + 2048*x^2 - 4096*x + 4096);

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

$C_2^3\times A_4$ (as 24T135):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

A solvable group of order 96

The 32 conjugacy class representatives for $C_2^3\times A_4$

Character table for $C_2^3\times A_4$ is not computed

Intermediate fields

$\Q(\sqrt{21}) $, $\Q(\sqrt{-3}) $, $\Q(\sqrt{-7}) $, $\Q(\zeta_{7})^+$, $\Q(\sqrt{-3}, \sqrt{-7})$, 6.0.64827.1, 6.6.4016873.1, 6.0.573839.1, $\Q(\zeta_{21})^+$, 6.0.108455571.1, 6.6.15493653.1, $\Q(\zeta_{7})$, 12.0.11762610880936041.3, 12.0.11762610880936041.1, 12.0.11762610880936041.2, 12.0.240053283284409.1, 12.12.11762610880936041.1, 12.0.16135268698129.1, $\Q(\zeta_{21})$

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

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

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

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

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

Sibling fields

Degree 24 siblings:	data not computed
Degree 32 sibling:	data not computed

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.6.0.1}{6} }^{4}$	R	${\href{/padicField/5.6.0.1}{6} }^{4}$	R	${\href{/padicField/11.6.0.1}{6} }^{4}$	${\href{/padicField/13.2.0.1}{2} }^{12}$	${\href{/padicField/17.6.0.1}{6} }^{4}$	${\href{/padicField/19.6.0.1}{6} }^{4}$	${\href{/padicField/23.6.0.1}{6} }^{4}$	${\href{/padicField/29.2.0.1}{2} }^{12}$	${\href{/padicField/31.6.0.1}{6} }^{4}$	${\href{/padicField/37.6.0.1}{6} }^{4}$	${\href{/padicField/41.2.0.1}{2} }^{12}$	${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{16}$	${\href{/padicField/47.6.0.1}{6} }^{4}$	${\href{/padicField/53.6.0.1}{6} }^{4}$	${\href{/padicField/59.6.0.1}{6} }^{4}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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
$3$ 3	3.12.6.2	$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
$3$ 3	3.12.6.2	$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
$7$ 7	7.12.10.1	$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$	$6$	$2$	$10$	$C_6\times C_2$	$[\ ]_{6}^{2}$
$7$ 7	7.12.10.1	$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$	$6$	$2$	$10$	$C_6\times C_2$	$[\ ]_{6}^{2}$
$239$ 239		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $4$	$2$	$2$	$2$
		Deg $4$	$2$	$2$	$2$

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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