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Label Class Base field Conductor norm Rank Torsion CM Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
7.1-a1 7.1-a 4.4.1957.1 \( 7 \) $0$ $\Z/2\Z$ $1$ $79.35083879$ 0.896863012 \( \frac{443958861593546408990}{117649} a^{3} + \frac{915220682926034165096}{117649} a^{2} + \frac{110890939600298461445}{117649} a - \frac{215357317054416491281}{117649} \) \( \bigl[a^{2} - a - 1\) , \( -a^{3} + a^{2} + 3 a - 2\) , \( a^{2} - a - 1\) , \( 32 a^{3} - 57 a^{2} - 33 a + 17\) , \( 204 a^{3} - 364 a^{2} - 182 a + 118\bigr] \) ${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-2\right){x}^{2}+\left(32a^{3}-57a^{2}-33a+17\right){x}+204a^{3}-364a^{2}-182a+118$
7.1-a2 7.1-a 4.4.1957.1 \( 7 \) $0$ $\Z/2\Z$ $1$ $158.7016775$ 0.896863012 \( \frac{124820}{7} a^{3} - \frac{238513}{7} a^{2} - \frac{74080}{7} a + \frac{71744}{7} \) \( \bigl[-a^{3} + a^{2} + 3 a\) , \( a^{3} - 5 a\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( a^{2} - 3 a\) , \( -a^{3} + a^{2} + a\bigr] \) ${y}^2+\left(-a^{3}+a^{2}+3a\right){x}{y}+\left(-a^{3}+a^{2}+3a-1\right){y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(a^{2}-3a\right){x}-a^{3}+a^{2}+a$
7.1-a3 7.1-a 4.4.1957.1 \( 7 \) $0$ $\Z/2\Z$ $1$ $79.35083879$ 0.896863012 \( -\frac{170631953648}{49} a^{3} + \frac{301490635046}{49} a^{2} + \frac{150363436017}{49} a - \frac{96330872074}{49} \) \( \bigl[-a^{3} + a^{2} + 3 a\) , \( a^{3} - 5 a\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( -5 a^{3} - 4 a^{2} + 7 a + 5\) , \( -6 a^{3} - 8 a^{2} - 3\bigr] \) ${y}^2+\left(-a^{3}+a^{2}+3a\right){x}{y}+\left(-a^{3}+a^{2}+3a-1\right){y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(-5a^{3}-4a^{2}+7a+5\right){x}-6a^{3}-8a^{2}-3$
7.1-a4 7.1-a 4.4.1957.1 \( 7 \) $0$ $\Z/2\Z$ $1$ $158.7016775$ 0.896863012 \( -\frac{9690935416}{343} a^{3} - \frac{12215671997}{343} a^{2} + \frac{16657874830}{343} a + \frac{14201869411}{343} \) \( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{3} - 3 a\) , \( -a^{3} - 4 a^{2} - 6 a + 1\) , \( -5 a^{3} - 11 a^{2} - a + 3\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{2}+a+1\right){x}^{2}+\left(-a^{3}-4a^{2}-6a+1\right){x}-5a^{3}-11a^{2}-a+3$
19.1-a1 19.1-a 4.4.1957.1 \( 19 \) $0$ $\mathsf{trivial}$ $1$ $61.31233619$ 1.385965601 \( -\frac{2062477872}{19} a^{3} - \frac{4353909970}{19} a^{2} - \frac{545558007}{19} a + \frac{1028555383}{19} \) \( \bigl[-a^{3} + a^{2} + 4 a - 1\) , \( -a^{3} + a^{2} + 3 a\) , \( a^{3} - 3 a - 1\) , \( 10 a^{3} - 17 a^{2} - 10 a + 7\) , \( 47 a^{3} - 82 a^{2} - 43 a + 26\bigr] \) ${y}^2+\left(-a^{3}+a^{2}+4a-1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a\right){x}^{2}+\left(10a^{3}-17a^{2}-10a+7\right){x}+47a^{3}-82a^{2}-43a+26$
19.1-b1 19.1-b 4.4.1957.1 \( 19 \) $0$ $\mathsf{trivial}$ $1$ $1.178437935$ 0.665964854 \( \frac{22217245546666612500}{6859} a^{3} + \frac{8805551373160716923}{6859} a^{2} - \frac{85379003522040468141}{6859} a - \frac{56056237584037687000}{6859} \) \( \bigl[a^{2} - 1\) , \( -2 a^{3} + a^{2} + 7 a - 1\) , \( -a^{3} + a^{2} + 3 a\) , \( 38 a^{3} - 44 a^{2} - 62 a - 20\) , \( 401 a^{3} - 614 a^{2} - 483 a + 89\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(-a^{3}+a^{2}+3a\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+7a-1\right){x}^{2}+\left(38a^{3}-44a^{2}-62a-20\right){x}+401a^{3}-614a^{2}-483a+89$
19.1-b2 19.1-b 4.4.1957.1 \( 19 \) $0$ $\Z/5\Z$ $1$ $736.5237097$ 0.665964854 \( \frac{10334}{19} a^{3} - \frac{3313}{19} a^{2} - \frac{36107}{19} a - \frac{5371}{19} \) \( \bigl[a^{2} - 1\) , \( -2 a^{3} + a^{2} + 7 a - 1\) , \( -a^{3} + a^{2} + 3 a\) , \( 3 a^{3} - 4 a^{2} - 7 a + 5\) , \( -3 a^{3} + 4 a^{2} + 6 a - 3\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(-a^{3}+a^{2}+3a\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+7a-1\right){x}^{2}+\left(3a^{3}-4a^{2}-7a+5\right){x}-3a^{3}+4a^{2}+6a-3$
19.1-b3 19.1-b 4.4.1957.1 \( 19 \) $0$ $\Z/5\Z$ $1$ $736.5237097$ 0.665964854 \( \frac{44732636264}{361} a^{3} - \frac{78909304839}{361} a^{2} - \frac{39733288819}{361} a + \frac{25358512200}{361} \) \( \bigl[-a^{3} + a^{2} + 4 a - 1\) , \( -2 a^{3} + a^{2} + 7 a - 1\) , \( a^{2} - a - 1\) , \( -a^{3} + 7 a + 6\) , \( -5 a^{3} - 7 a^{2} + 11 a + 10\bigr] \) ${y}^2+\left(-a^{3}+a^{2}+4a-1\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+7a-1\right){x}^{2}+\left(-a^{3}+7a+6\right){x}-5a^{3}-7a^{2}+11a+10$
19.1-b4 19.1-b 4.4.1957.1 \( 19 \) $0$ $\mathsf{trivial}$ $1$ $1.178437935$ 0.665964854 \( -\frac{7261286418859671754}{16983563041} a^{3} - \frac{14970804000790893451}{16983563041} a^{2} - \frac{1817822990455263041}{16983563041} a + \frac{3520277452889246795}{16983563041} \) \( \bigl[1\) , \( -a^{3} + 5 a\) , \( a^{2} - 2\) , \( -38 a^{3} + 67 a^{2} + 92 a - 143\) , \( -658 a^{3} + 655 a^{2} + 2105 a - 1361\bigr] \) ${y}^2+{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+\left(-38a^{3}+67a^{2}+92a-143\right){x}-658a^{3}+655a^{2}+2105a-1361$
21.1-a1 21.1-a 4.4.1957.1 \( 3 \cdot 7 \) $0$ $\Z/8\Z$ $1$ $1050.985704$ 0.742422999 \( -\frac{4584696391003}{250047} a^{3} - \frac{1823709580711}{250047} a^{2} + \frac{5875898465972}{83349} a + \frac{11577122282035}{250047} \) \( \bigl[a^{2} - 1\) , \( a^{3} - a^{2} - 2 a + 1\) , \( a^{3} - 3 a - 1\) , \( 2 a^{3} - a^{2} - 5 a\) , \( 7 a^{3} - 2 a^{2} - 30 a + 11\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+1\right){x}^{2}+\left(2a^{3}-a^{2}-5a\right){x}+7a^{3}-2a^{2}-30a+11$
21.1-a2 21.1-a 4.4.1957.1 \( 3 \cdot 7 \) $0$ $\Z/2\Z$ $1$ $2.052706454$ 0.742422999 \( -\frac{91305470986913768828360282696367059}{9384442261550973912914558289} a^{3} - \frac{36191380838537641287089198334152081}{9384442261550973912914558289} a^{2} + \frac{116960169615559391050979984902302073}{3128147420516991304304852763} a + \frac{230385512085604356994786103257239902}{9384442261550973912914558289} \) \( \bigl[a^{2} - 1\) , \( a^{3} - a^{2} - 2 a + 1\) , \( a^{3} - 3 a - 1\) , \( 312 a^{3} - 201 a^{2} - 1090 a + 375\) , \( -2482 a^{3} + 1995 a^{2} + 8193 a - 3631\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+1\right){x}^{2}+\left(312a^{3}-201a^{2}-1090a+375\right){x}-2482a^{3}+1995a^{2}+8193a-3631$
21.1-a3 21.1-a 4.4.1957.1 \( 3 \cdot 7 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $32.84330326$ 0.742422999 \( -\frac{49478337663774144106792217}{3909188328478827879681} a^{3} + \frac{86867577756332942729622262}{3909188328478827879681} a^{2} + \frac{19815761443619939196843979}{1303062776159609293227} a - \frac{6943686789535186694527711}{3909188328478827879681} \) \( \bigl[-a^{3} + a^{2} + 3 a - 1\) , \( -a^{3} + a^{2} + 2 a - 1\) , \( a^{3} - 3 a - 1\) , \( -14 a^{3} - 52 a^{2} - 24 a + 2\) , \( -149 a^{3} - 293 a^{2} - 54 a + 53\bigr] \) ${y}^2+\left(-a^{3}+a^{2}+3a-1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a-1\right){x}^{2}+\left(-14a^{3}-52a^{2}-24a+2\right){x}-149a^{3}-293a^{2}-54a+53$
21.1-a4 21.1-a 4.4.1957.1 \( 3 \cdot 7 \) $0$ $\Z/2\Z\oplus\Z/4\Z$ $1$ $525.4928522$ 0.742422999 \( -\frac{84413562014875846901}{62523502209} a^{3} + \frac{148907375867724857512}{62523502209} a^{2} + \frac{24992949793645640572}{20841167403} a - \frac{47851964758998446149}{62523502209} \) \( \bigl[-a^{3} + a^{2} + 3 a - 1\) , \( -a^{3} + a^{2} + 2 a - 1\) , \( a^{3} - 3 a - 1\) , \( a^{3} - 17 a^{2} - 9 a + 2\) , \( 8 a^{3} + 41 a^{2} + 11 a - 9\bigr] \) ${y}^2+\left(-a^{3}+a^{2}+3a-1\right){x}{y}+\left(a^{3}-3a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a-1\right){x}^{2}+\left(a^{3}-17a^{2}-9a+2\right){x}+8a^{3}+41a^{2}+11a-9$
21.1-a5 21.1-a 4.4.1957.1 \( 3 \cdot 7 \) $0$ $\Z/2\Z$ $1$ $2.052706454$ 0.742422999 \( -\frac{29879008921030774408202450507522}{2109289329} a^{3} - \frac{11842202503929076694840862982208}{2109289329} a^{2} + \frac{38274171513246722984375757845464}{703096443} a + \frac{75387595661097076378967854941161}{2109289329} \) \( \bigl[a\) , \( a^{2} - a - 2\) , \( a^{3} - 3 a\) , \( -12 a^{3} + 21 a^{2} - 3 a - 34\) , \( 660 a^{3} - 1000 a^{2} - 561 a + 285\bigr] \) ${y}^2+a{x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(-12a^{3}+21a^{2}-3a-34\right){x}+660a^{3}-1000a^{2}-561a+285$
21.1-a6 21.1-a 4.4.1957.1 \( 3 \cdot 7 \) $0$ $\Z/4\Z$ $1$ $262.7464261$ 0.742422999 \( -\frac{4031277404041470845619545}{250047} a^{3} + \frac{7111233508460767828860022}{250047} a^{2} + \frac{1193595856869520566460555}{83349} a - \frac{2285285314923636665330287}{250047} \) \( \bigl[-a^{3} + a^{2} + 3 a\) , \( -1\) , \( a + 1\) , \( -80 a^{3} - 32 a^{2} + 311 a + 126\) , \( -989 a^{3} - 406 a^{2} + 3795 a + 2783\bigr] \) ${y}^2+\left(-a^{3}+a^{2}+3a\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-80a^{3}-32a^{2}+311a+126\right){x}-989a^{3}-406a^{2}+3795a+2783$
21.1-a7 21.1-a 4.4.1957.1 \( 3 \cdot 7 \) $0$ $\Z/2\Z$ $1$ $2.052706454$ 0.742422999 \( \frac{981621700851969745102419250048115}{101814121186119595835681841} a^{3} + \frac{1761760795662379320695876261373041}{101814121186119595835681841} a^{2} - \frac{37671995275184386601460285791065}{33938040395373198611893947} a - \frac{293036539712498872630739037955790}{101814121186119595835681841} \) \( \bigl[-a^{3} + a^{2} + 3 a\) , \( -1\) , \( a + 1\) , \( -355 a^{3} - 122 a^{2} + 1271 a + 666\) , \( 7906 a^{3} + 3150 a^{2} - 31007 a - 21110\bigr] \) ${y}^2+\left(-a^{3}+a^{2}+3a\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-355a^{3}-122a^{2}+1271a+666\right){x}+7906a^{3}+3150a^{2}-31007a-21110$
21.1-a8 21.1-a 4.4.1957.1 \( 3 \cdot 7 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $32.84330326$ 0.742422999 \( -\frac{921288230848937984}{15752961} a^{3} - \frac{392913302754752516}{15752961} a^{2} + \frac{1182898550569873408}{5250987} a + \frac{2425491268899001073}{15752961} \) \( \bigl[1\) , \( -a^{2} + 2\) , \( 0\) , \( 20 a^{2} - 89\) , \( 100 a^{2} - 16 a - 384\bigr] \) ${y}^2+{x}{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(20a^{2}-89\right){x}+100a^{2}-16a-384$
21.1-a9 21.1-a 4.4.1957.1 \( 3 \cdot 7 \) $0$ $\Z/2\Z\oplus\Z/4\Z$ $1$ $525.4928522$ 0.742422999 \( \frac{18710001614848}{3969} a^{3} + \frac{38571223778296}{3969} a^{2} + \frac{1558338097840}{1323} a - \frac{9074746748287}{3969} \) \( \bigl[1\) , \( -a^{2} + 2\) , \( 0\) , \( -4\) , \( 4 a^{2} - 9\bigr] \) ${y}^2+{x}{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}-4{x}+4a^{2}-9$
21.1-a10 21.1-a 4.4.1957.1 \( 3 \cdot 7 \) $0$ $\Z/4\Z$ $1$ $262.7464261$ 0.742422999 \( \frac{24573261493292438142976}{63} a^{3} + \frac{50657750325995678467324}{63} a^{2} + \frac{2045949367379882389504}{21} a - \frac{11920094685054769076767}{63} \) \( \bigl[1\) , \( -a^{2} + 2\) , \( 0\) , \( -20 a^{2} + 1\) , \( 84 a^{2} + 16 a - 30\bigr] \) ${y}^2+{x}{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-20a^{2}+1\right){x}+84a^{2}+16a-30$
21.1-a11 21.1-a 4.4.1957.1 \( 3 \cdot 7 \) $0$ $\Z/8\Z$ $1$ $1050.985704$ 0.742422999 \( -\frac{1064704}{63} a^{3} - \frac{2122864}{63} a^{2} - \frac{47392}{21} a + \frac{567631}{63} \) \( \bigl[1\) , \( -a^{2} + 2\) , \( 0\) , \( 1\) , \( 0\bigr] \) ${y}^2+{x}{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+{x}$
21.1-a12 21.1-a 4.4.1957.1 \( 3 \cdot 7 \) $0$ $\Z/2\Z$ $1$ $2.052706454$ 0.742422999 \( \frac{3925517460083619669474871874}{466948881} a^{3} - \frac{2723612167164915292630390720}{466948881} a^{2} - \frac{4604122185339962156855866072}{155649627} a + \frac{5657812633917689859889126471}{466948881} \) \( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 2\) , \( 59 a^{3} - 73 a^{2} - 337 a - 189\) , \( 583 a^{3} - 32 a^{2} - 2275 a - 1390\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(59a^{3}-73a^{2}-337a-189\right){x}+583a^{3}-32a^{2}-2275a-1390$
23.1-a1 23.1-a 4.4.1957.1 \( 23 \) $0$ $\Z/2\Z$ $1$ $84.90966535$ 0.959691660 \( -\frac{7418820639147057}{529} a^{3} - \frac{2940364437579043}{529} a^{2} + \frac{28509902770920202}{529} a + \frac{18718393590733878}{529} \) \( \bigl[a^{3} - 4 a\) , \( a^{3} - 5 a - 2\) , \( a^{2} - 2\) , \( 6 a^{3} - 4 a^{2} - 31 a - 9\) , \( 10 a^{3} + 14 a^{2} - 16 a - 17\bigr] \) ${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(6a^{3}-4a^{2}-31a-9\right){x}+10a^{3}+14a^{2}-16a-17$
23.1-a2 23.1-a 4.4.1957.1 \( 23 \) $0$ $\Z/2\Z$ $1$ $169.8193307$ 0.959691660 \( \frac{43532247}{23} a^{3} + \frac{17211519}{23} a^{2} - \frac{167270339}{23} a - \frac{109671331}{23} \) \( \bigl[a^{3} - 3 a\) , \( -2 a^{3} + a^{2} + 8 a + 1\) , \( a^{2} - a - 1\) , \( 3 a + 4\) , \( a^{3} - a^{2} + a + 1\bigr] \) ${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+8a+1\right){x}^{2}+\left(3a+4\right){x}+a^{3}-a^{2}+a+1$
31.1-a1 31.1-a 4.4.1957.1 \( 31 \) $1$ $\mathsf{trivial}$ $1.022761215$ $2.399984655$ 1.109729927 \( -\frac{180825454881646489600000}{28629151} a^{3} + \frac{318978801202372556800000}{28629151} a^{2} + \frac{160618453429727667920896}{28629151} a - \frac{102507894005169170706432}{28629151} \) \( \bigl[0\) , \( a^{3} - 4 a - 1\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( 69 a^{3} + 29 a^{2} - 257 a - 177\) , \( 387 a^{3} + 140 a^{2} - 1515 a - 1004\bigr] \) ${y}^2+\left(-a^{3}+a^{2}+3a-1\right){y}={x}^{3}+\left(a^{3}-4a-1\right){x}^{2}+\left(69a^{3}+29a^{2}-257a-177\right){x}+387a^{3}+140a^{2}-1515a-1004$
31.1-a2 31.1-a 4.4.1957.1 \( 31 \) $1$ $\Z/5\Z$ $0.068184081$ $1499.990409$ 1.109729927 \( \frac{446038016}{29791} a^{3} + \frac{34282332160}{29791} a^{2} - \frac{48219545600}{29791} a - \frac{50153631744}{29791} \) \( \bigl[0\) , \( a^{3} - 5 a\) , \( -a^{3} + a^{2} + 4 a\) , \( 2 a^{3} + 2 a^{2} - 8 a - 6\) , \( -5 a^{3} - 2 a^{2} + 18 a + 11\bigr] \) ${y}^2+\left(-a^{3}+a^{2}+4a\right){y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(2a^{3}+2a^{2}-8a-6\right){x}-5a^{3}-2a^{2}+18a+11$
31.1-a3 31.1-a 4.4.1957.1 \( 31 \) $1$ $\mathsf{trivial}$ $0.340920405$ $2.399984655$ 1.109729927 \( -\frac{2536784722881790079590400000}{23465261991844685929951} a^{3} - \frac{1326320955838364685202235392}{23465261991844685929951} a^{2} + \frac{561810697949918688555835392}{23465261991844685929951} a + \frac{181027024945269059059658752}{23465261991844685929951} \) \( \bigl[0\) , \( a^{3} - 5 a\) , \( -a^{3} + a^{2} + 4 a\) , \( -28 a^{3} - 38 a^{2} + 52 a + 54\) , \( -103 a^{3} - 116 a^{2} + 64 a - 103\bigr] \) ${y}^2+\left(-a^{3}+a^{2}+4a\right){y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(-28a^{3}-38a^{2}+52a+54\right){x}-103a^{3}-116a^{2}+64a-103$
31.1-a4 31.1-a 4.4.1957.1 \( 31 \) $1$ $\Z/5\Z$ $0.204552243$ $1499.990409$ 1.109729927 \( -\frac{65536}{31} a^{3} - \frac{139264}{31} a^{2} - \frac{65536}{31} a + \frac{53248}{31} \) \( \bigl[0\) , \( -a^{3} + 3 a\) , \( 1\) , \( 3 a^{3} - 2 a^{2} - 10 a + 5\) , \( -2 a^{3} + a^{2} + 7 a - 2\bigr] \) ${y}^2+{y}={x}^{3}+\left(-a^{3}+3a\right){x}^{2}+\left(3a^{3}-2a^{2}-10a+5\right){x}-2a^{3}+a^{2}+7a-2$
43.1-a1 43.1-a 4.4.1957.1 \( 43 \) $1$ $\mathsf{trivial}$ $0.006791259$ $669.3900782$ 1.233148718 \( \frac{1592683107521}{79507} a^{3} + \frac{629635995686}{79507} a^{2} - \frac{6120072768045}{79507} a - \frac{4012649003175}{79507} \) \( \bigl[a^{2} - 1\) , \( 2 a^{3} - a^{2} - 8 a\) , \( 0\) , \( -3 a^{3} + 3 a^{2} + 10 a - 7\) , \( -a^{3} - 2 a^{2} + 4 a + 9\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}={x}^{3}+\left(2a^{3}-a^{2}-8a\right){x}^{2}+\left(-3a^{3}+3a^{2}+10a-7\right){x}-a^{3}-2a^{2}+4a+9$
43.1-a2 43.1-a 4.4.1957.1 \( 43 \) $1$ $\mathsf{trivial}$ $0.020373779$ $669.3900782$ 1.233148718 \( \frac{655988}{43} a^{3} - \frac{455074}{43} a^{2} - \frac{2306280}{43} a + \frac{943113}{43} \) \( \bigl[1\) , \( -a - 1\) , \( a^{2} - a - 1\) , \( a\) , \( -a^{2} + a + 1\bigr] \) ${y}^2+{x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+a{x}-a^{2}+a+1$
47.1-a1 47.1-a 4.4.1957.1 \( 47 \) $0$ $\Z/2\Z$ $1$ $151.5709415$ 1.713130865 \( \frac{26469954681304186}{2209} a^{3} - \frac{18365448892640224}{2209} a^{2} - \frac{93137457700655592}{2209} a + \frac{38150905028515907}{2209} \) \( \bigl[a^{2} - 2\) , \( a\) , \( a^{2} - a - 2\) , \( -a^{2} + 3 a\) , \( 64 a^{3} + 28 a^{2} - 249 a - 166\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+a{x}^{2}+\left(-a^{2}+3a\right){x}+64a^{3}+28a^{2}-249a-166$
47.1-a2 47.1-a 4.4.1957.1 \( 47 \) $0$ $\Z/2\Z$ $1$ $303.1418830$ 1.713130865 \( \frac{1184609061189}{103823} a^{3} + \frac{2580667051169}{103823} a^{2} + \frac{343044349773}{103823} a - \frac{608636026432}{103823} \) \( \bigl[a^{2} - a - 1\) , \( -2 a^{3} + a^{2} + 8 a\) , \( -a^{3} + a^{2} + 4 a - 1\) , \( 7 a^{3} - 3 a^{2} - 21 a - 4\) , \( -5 a^{3} + 2 a^{2} + 15 a + 2\bigr] \) ${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(-a^{3}+a^{2}+4a-1\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+8a\right){x}^{2}+\left(7a^{3}-3a^{2}-21a-4\right){x}-5a^{3}+2a^{2}+15a+2$
47.1-a3 47.1-a 4.4.1957.1 \( 47 \) $0$ $\Z/2\Z$ $1$ $151.5709415$ 1.713130865 \( -\frac{11148830379801159533743}{10779215329} a^{3} + \frac{19666703347863657066005}{10779215329} a^{2} + \frac{9902963271520642369814}{10779215329} a - \frac{6320145061844672607397}{10779215329} \) \( \bigl[a^{2} - a - 1\) , \( -2 a^{3} + a^{2} + 8 a\) , \( -a^{3} + a^{2} + 4 a - 1\) , \( 32 a^{3} - 58 a^{2} - 26 a + 21\) , \( -197 a^{3} + 305 a^{2} + 233 a - 53\bigr] \) ${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(-a^{3}+a^{2}+4a-1\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+8a\right){x}^{2}+\left(32a^{3}-58a^{2}-26a+21\right){x}-197a^{3}+305a^{2}+233a-53$
47.1-a4 47.1-a 4.4.1957.1 \( 47 \) $0$ $\Z/2\Z$ $1$ $303.1418830$ 1.713130865 \( -\frac{91309792}{47} a^{3} + \frac{62772516}{47} a^{2} + \frac{321433088}{47} a - \frac{129364665}{47} \) \( \bigl[a^{3} - 4 a\) , \( -2 a^{3} + a^{2} + 7 a - 1\) , \( a^{3} - 4 a - 1\) , \( 2 a^{3} - 3 a^{2} - 4 a + 1\) , \( 2 a^{3} - 3 a^{2} - 2 a - 1\bigr] \) ${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+7a-1\right){x}^{2}+\left(2a^{3}-3a^{2}-4a+1\right){x}+2a^{3}-3a^{2}-2a-1$
47.1-b1 47.1-b 4.4.1957.1 \( 47 \) $0 \le r \le 2$ $\Z/4\Z$ $1$ $690.3504462$ 0.975335910 \( -\frac{1527277}{47} a^{3} + \frac{1688549}{47} a^{2} + \frac{2970352}{47} a + \frac{259372}{47} \) \( \bigl[a^{2} - 2\) , \( -2 a^{3} + a^{2} + 8 a + 1\) , \( -a^{3} + a^{2} + 3 a\) , \( -3 a^{3} - 4 a^{2} + 7 a + 7\) , \( -a^{3} - a^{2} + 4 a + 2\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(-a^{3}+a^{2}+3a\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+8a+1\right){x}^{2}+\left(-3a^{3}-4a^{2}+7a+7\right){x}-a^{3}-a^{2}+4a+2$
47.1-b2 47.1-b 4.4.1957.1 \( 47 \) $0 \le r \le 2$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $86.29380577$ 0.975335910 \( -\frac{8324109193351}{2209} a^{3} + \frac{14771368925300}{2209} a^{2} + \frac{7178823816030}{2209} a - \frac{4647340584154}{2209} \) \( \bigl[a^{2} - 2\) , \( -2 a^{3} + a^{2} + 8 a + 1\) , \( -a^{3} + a^{2} + 3 a\) , \( -13 a^{3} - 19 a^{2} + 12 a + 7\) , \( -65 a^{3} - 133 a^{2} - 13 a + 31\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(-a^{3}+a^{2}+3a\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+8a+1\right){x}^{2}+\left(-13a^{3}-19a^{2}+12a+7\right){x}-65a^{3}-133a^{2}-13a+31$
47.1-b3 47.1-b 4.4.1957.1 \( 47 \) $0 \le r \le 2$ $\Z/2\Z$ $1$ $5.393362860$ 0.975335910 \( \frac{21493599447952576879551}{4879681} a^{3} - \frac{14912741973407868587654}{4879681} a^{2} - \frac{75627602524472124243387}{4879681} a + \frac{30978529535192993029238}{4879681} \) \( \bigl[a^{2} - 1\) , \( a^{3} - 4 a\) , \( a^{3} - 4 a - 1\) , \( 49 a^{3} - 80 a^{2} - 46 a + 12\) , \( 423 a^{3} - 748 a^{2} - 365 a + 218\bigr] \) ${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(a^{3}-4a\right){x}^{2}+\left(49a^{3}-80a^{2}-46a+12\right){x}+423a^{3}-748a^{2}-365a+218$
47.1-b4 47.1-b 4.4.1957.1 \( 47 \) $0 \le r \le 2$ $\Z/2\Z$ $1$ $10.78672572$ 0.975335910 \( -\frac{6019345658438801329873}{47} a^{3} + \frac{10618215581395079746074}{47} a^{2} + \frac{5346691869248498657045}{47} a - \frac{3412298599331115695098}{47} \) \( \bigl[a\) , \( -a^{3} + a^{2} + 2 a\) , \( -a^{3} + a^{2} + 3 a\) , \( -75 a^{3} - 29 a^{2} + 245 a + 124\) , \( 171 a^{3} + 21 a^{2} - 837 a - 601\bigr] \) ${y}^2+a{x}{y}+\left(-a^{3}+a^{2}+3a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a\right){x}^{2}+\left(-75a^{3}-29a^{2}+245a+124\right){x}+171a^{3}+21a^{2}-837a-601$
47.1-c1 47.1-c 4.4.1957.1 \( 47 \) $1$ $\Z/2\Z$ $0.030350985$ $805.5519662$ 1.105352853 \( \frac{260493073}{2209} a^{3} - \frac{513538831655}{2209} a^{2} + \frac{153993847068}{2209} a + \frac{1871080196928}{2209} \) \( \bigl[a^{2} - a - 2\) , \( 2 a^{3} - a^{2} - 8 a + 1\) , \( -a^{3} + a^{2} + 3 a\) , \( -13 a^{3} + 8 a^{2} + 44 a - 21\) , \( 16 a^{3} - 14 a^{2} - 57 a + 29\bigr] \) ${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(-a^{3}+a^{2}+3a\right){y}={x}^{3}+\left(2a^{3}-a^{2}-8a+1\right){x}^{2}+\left(-13a^{3}+8a^{2}+44a-21\right){x}+16a^{3}-14a^{2}-57a+29$
47.1-c2 47.1-c 4.4.1957.1 \( 47 \) $1$ $\Z/2\Z$ $0.015175492$ $3222.207864$ 1.105352853 \( -\frac{5774625}{47} a^{3} + \frac{9047691}{47} a^{2} + \frac{7169077}{47} a - \frac{2507139}{47} \) \( \bigl[a^{2} - a - 2\) , \( 2 a^{3} - a^{2} - 8 a + 1\) , \( -a^{3} + a^{2} + 3 a\) , \( 2 a^{3} - 2 a^{2} - 6 a + 4\) , \( a^{3} - a^{2} - 3 a + 1\bigr] \) ${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(-a^{3}+a^{2}+3a\right){y}={x}^{3}+\left(2a^{3}-a^{2}-8a+1\right){x}^{2}+\left(2a^{3}-2a^{2}-6a+4\right){x}+a^{3}-a^{2}-3a+1$
48.1-a1 48.1-a 4.4.1957.1 \( 2^{4} \cdot 3 \) $0$ $\Z/5\Z$ $1$ $319.8556448$ 1.446067622 \( -\frac{256787}{486} a^{3} - \frac{917825}{486} a^{2} - \frac{221027}{162} a + \frac{155944}{243} \) \( \bigl[-a^{3} + a^{2} + 3 a - 1\) , \( -a^{3} + 5 a + 2\) , \( -a^{3} + a^{2} + 4 a\) , \( -2 a^{3} + a^{2} + 7 a + 2\) , \( 2 a^{3} - 5 a^{2} + 2\bigr] \) ${y}^2+\left(-a^{3}+a^{2}+3a-1\right){x}{y}+\left(-a^{3}+a^{2}+4a\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(-2a^{3}+a^{2}+7a+2\right){x}+2a^{3}-5a^{2}+2$
48.1-a2 48.1-a 4.4.1957.1 \( 2^{4} \cdot 3 \) $0$ $\mathsf{trivial}$ $1$ $0.511769031$ 1.446067622 \( \frac{2808717352846811}{48} a^{3} + \frac{2225925010621747}{96} a^{2} - \frac{7195553538165561}{32} a - \frac{14172658068106189}{96} \) \( \bigl[-a^{3} + a^{2} + 4 a\) , \( -a^{2} + 2 a + 2\) , \( a^{2} - a - 2\) , \( -255 a^{3} + 230 a^{2} + 886 a - 555\) , \( -3441 a^{3} + 2714 a^{2} + 11994 a - 6178\bigr] \) ${y}^2+\left(-a^{3}+a^{2}+4a\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(-255a^{3}+230a^{2}+886a-555\right){x}-3441a^{3}+2714a^{2}+11994a-6178$
48.1-b1 48.1-b 4.4.1957.1 \( 2^{4} \cdot 3 \) $0$ $\mathsf{trivial}$ $1$ $41.70561828$ 0.942755665 \( -\frac{4778497131611}{157464} a^{3} - \frac{1148773643641}{78732} a^{2} + \frac{761381050117}{13122} a - \frac{370433033072}{19683} \) \( \bigl[-a^{3} + a^{2} + 3 a\) , \( a^{3} - 4 a - 1\) , \( a^{2} - 1\) , \( -6 a^{3} + a^{2} + 13 a - 4\) , \( -3 a^{3} - 2 a^{2} - 7 a + 3\bigr] \) ${y}^2+\left(-a^{3}+a^{2}+3a\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-4a-1\right){x}^{2}+\left(-6a^{3}+a^{2}+13a-4\right){x}-3a^{3}-2a^{2}-7a+3$
48.1-b2 48.1-b 4.4.1957.1 \( 2^{4} \cdot 3 \) $0$ $\mathsf{trivial}$ $1$ $41.70561828$ 0.942755665 \( \frac{938631780020}{27} a^{3} + \frac{372015936188}{27} a^{2} - \frac{2404721908865}{18} a - \frac{2368257700055}{27} \) \( \bigl[1\) , \( a^{3} - 4 a - 2\) , \( a^{2} - a - 2\) , \( 6 a^{3} - 13 a^{2} - 2 a + 7\) , \( 40 a^{3} - 69 a^{2} - 38 a + 21\bigr] \) ${y}^2+{x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{3}-4a-2\right){x}^{2}+\left(6a^{3}-13a^{2}-2a+7\right){x}+40a^{3}-69a^{2}-38a+21$
49.1-a1 49.1-a 4.4.1957.1 \( 7^{2} \) $1$ $\Z/2\Z$ $0.029401742$ $432.7430195$ 1.150449917 \( -\frac{9690935416}{343} a^{3} - \frac{12215671997}{343} a^{2} + \frac{16657874830}{343} a + \frac{14201869411}{343} \) \( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 4 a - 1\) , \( a\) , \( -30 a^{3} + 55 a^{2} + 22 a - 15\) , \( -267 a^{3} + 469 a^{2} + 242 a - 153\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+a^{2}+4a-1\right){x}^{2}+\left(-30a^{3}+55a^{2}+22a-15\right){x}-267a^{3}+469a^{2}+242a-153$
49.1-a2 49.1-a 4.4.1957.1 \( 7^{2} \) $1$ $\Z/2\Z$ $0.019601161$ $649.1145293$ 1.150449917 \( -\frac{170631953648}{49} a^{3} + \frac{301490635046}{49} a^{2} + \frac{150363436017}{49} a - \frac{96330872074}{49} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + a + 3\) , \( a^{3} - 4 a - 1\) , \( -63 a^{3} + 23 a^{2} + 227 a - 17\) , \( 162 a^{3} - 125 a^{2} - 566 a + 280\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(-63a^{3}+23a^{2}+227a-17\right){x}+162a^{3}-125a^{2}-566a+280$
49.1-a3 49.1-a 4.4.1957.1 \( 7^{2} \) $1$ $\Z/2\Z$ $0.009800580$ $1298.229058$ 1.150449917 \( \frac{124820}{7} a^{3} - \frac{238513}{7} a^{2} - \frac{74080}{7} a + \frac{71744}{7} \) \( \bigl[a^{3} - 4 a\) , \( a^{3} - 3 a\) , \( a^{2} - a - 1\) , \( a^{3} - a^{2} - 3 a\) , \( a^{3} - a\bigr] \) ${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(a^{3}-a^{2}-3a\right){x}+a^{3}-a$
49.1-a4 49.1-a 4.4.1957.1 \( 7^{2} \) $1$ $\Z/2\Z$ $0.058803484$ $216.3715097$ 1.150449917 \( \frac{443958861593546408990}{117649} a^{3} + \frac{915220682926034165096}{117649} a^{2} + \frac{110890939600298461445}{117649} a - \frac{215357317054416491281}{117649} \) \( \bigl[a\) , \( -a^{3} + 3 a + 1\) , \( -a^{3} + a^{2} + 4 a\) , \( -a^{3} - 16 a^{2} + 56 a - 44\) , \( -109 a^{3} + 141 a^{2} + 199 a - 57\bigr] \) ${y}^2+a{x}{y}+\left(-a^{3}+a^{2}+4a\right){y}={x}^{3}+\left(-a^{3}+3a+1\right){x}^{2}+\left(-a^{3}-16a^{2}+56a-44\right){x}-109a^{3}+141a^{2}+199a-57$
57.1-a1 57.1-a 4.4.1957.1 \( 3 \cdot 19 \) $0$ $\Z/2\Z$ $1$ $48.45722553$ 1.095375774 \( -\frac{1236877195737659}{1083} a^{3} - \frac{484396917322811}{1083} a^{2} + \frac{1580740068049001}{361} a + \frac{3118669260369488}{1083} \) \( \bigl[a^{2} - a - 1\) , \( -a^{3} + 5 a\) , \( 0\) , \( -59 a^{3} + 40 a^{2} + 158 a - 173\) , \( 730 a^{3} - 219 a^{2} - 2417 a + 422\bigr] \) ${y}^2+\left(a^{2}-a-1\right){x}{y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+\left(-59a^{3}+40a^{2}+158a-173\right){x}+730a^{3}-219a^{2}-2417a+422$
57.1-a2 57.1-a 4.4.1957.1 \( 3 \cdot 19 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $193.8289021$ 1.095375774 \( -\frac{32864500283}{61731} a^{3} - \frac{638130530}{3249} a^{2} + \frac{41046464113}{20577} a + \frac{85760775575}{61731} \) \( \bigl[a^{2} - a - 1\) , \( -a^{3} + 5 a\) , \( 0\) , \( -4 a^{3} + 8 a - 8\) , \( 12 a^{3} - 13 a^{2} - 54 a + 11\bigr] \) ${y}^2+\left(a^{2}-a-1\right){x}{y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+\left(-4a^{3}+8a-8\right){x}+12a^{3}-13a^{2}-54a+11$
57.1-a3 57.1-a 4.4.1957.1 \( 3 \cdot 19 \) $0$ $\Z/2\Z$ $1$ $48.45722553$ 1.095375774 \( \frac{72942210831600947693872817724388}{513} a^{3} - \frac{50608943902051469792355565222382}{513} a^{2} - \frac{85551740515727781170115992032406}{171} a + \frac{105130945456975830283712819159561}{513} \) \( \bigl[-a^{3} + a^{2} + 3 a - 1\) , \( a^{2} - 2 a - 1\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( 52 a^{3} + 99 a^{2} - 225 a - 448\) , \( 1490 a^{3} + 58 a^{2} - 5543 a - 1945\bigr] \) ${y}^2+\left(-a^{3}+a^{2}+3a-1\right){x}{y}+\left(-a^{3}+a^{2}+3a-1\right){y}={x}^{3}+\left(a^{2}-2a-1\right){x}^{2}+\left(52a^{3}+99a^{2}-225a-448\right){x}+1490a^{3}+58a^{2}-5543a-1945$
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  *The rank, regulator and analytic order of Ш are not known for all curves in the database; curves for which these are unknown will not appear in searches specifying one of these quantities.