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Label Class Base field Conductor norm Rank Torsion CM Sato-Tate Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
79.4-a1 79.4-a \(\Q(\zeta_{13})^+\) \( 79 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.019263609$ $53060.19998$ 2.51617 \( \frac{22488882826}{79} a^{5} - \frac{6539496375}{79} a^{4} - 1482055738 a^{3} + \frac{6919343568}{79} a^{2} + \frac{139840964971}{79} a + \frac{31710174030}{79} \) \( \bigl[a^{5} - 4 a^{3} + 2 a + 1\) , \( -a^{5} - a^{4} + 4 a^{3} + 5 a^{2} - a - 3\) , \( 1\) , \( -5 a^{5} - 2 a^{4} + 21 a^{3} + 11 a^{2} - 13 a - 3\) , \( -3 a^{5} + a^{4} + 14 a^{3} + 2 a^{2} - 7 a - 1\bigr] \) ${y}^2+\left(a^{5}-4a^{3}+2a+1\right){x}{y}+{y}={x}^{3}+\left(-a^{5}-a^{4}+4a^{3}+5a^{2}-a-3\right){x}^{2}+\left(-5a^{5}-2a^{4}+21a^{3}+11a^{2}-13a-3\right){x}-3a^{5}+a^{4}+14a^{3}+2a^{2}-7a-1$
79.4-a2 79.4-a \(\Q(\zeta_{13})^+\) \( 79 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.038527218$ $13265.04999$ 2.51617 \( \frac{2136052117235807108333}{6241} a^{5} - \frac{621143077737174532085}{6241} a^{4} - \frac{140769383898930680610}{79} a^{3} + \frac{657241655652391415182}{6241} a^{2} + \frac{13282434909539556190067}{6241} a + \frac{3011876309786523676503}{6241} \) \( \bigl[a^{5} - 4 a^{3} + 2 a + 1\) , \( -a^{5} - a^{4} + 4 a^{3} + 5 a^{2} - a - 3\) , \( 1\) , \( -20 a^{5} - 7 a^{4} + 71 a^{3} + 11 a^{2} - 33 a - 3\) , \( 31 a^{5} + 51 a^{4} - 97 a^{3} - 126 a^{2} + 78 a + 24\bigr] \) ${y}^2+\left(a^{5}-4a^{3}+2a+1\right){x}{y}+{y}={x}^{3}+\left(-a^{5}-a^{4}+4a^{3}+5a^{2}-a-3\right){x}^{2}+\left(-20a^{5}-7a^{4}+71a^{3}+11a^{2}-33a-3\right){x}+31a^{5}+51a^{4}-97a^{3}-126a^{2}+78a+24$
79.4-b1 79.4-b \(\Q(\zeta_{13})^+\) \( 79 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.001171369$ $107610.2235$ 2.48239 \( -\frac{36496668313}{6241} a^{5} + \frac{77850269004}{6241} a^{4} + \frac{1194357473}{79} a^{3} - \frac{253201872353}{6241} a^{2} + \frac{68224588757}{6241} a + \frac{32165821654}{6241} \) \( \bigl[1\) , \( -a^{3} + a^{2} + 2 a - 3\) , \( a^{5} + a^{4} - 4 a^{3} - 4 a^{2} + 3 a + 2\) , \( 4 a^{5} + 5 a^{4} - 19 a^{3} - 20 a^{2} + 14 a + 5\) , \( -29 a^{5} - 20 a^{4} + 127 a^{3} + 87 a^{2} - 82 a - 21\bigr] \) ${y}^2+{x}{y}+\left(a^{5}+a^{4}-4a^{3}-4a^{2}+3a+2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a-3\right){x}^{2}+\left(4a^{5}+5a^{4}-19a^{3}-20a^{2}+14a+5\right){x}-29a^{5}-20a^{4}+127a^{3}+87a^{2}-82a-21$
79.4-c1 79.4-c \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/10\Z$ $\mathrm{SU}(2)$ $1$ $45606.09482$ 1.49691 \( -\frac{506078693721616}{6241} a^{5} - \frac{251520007112219}{6241} a^{4} + \frac{27264105978348}{79} a^{3} + \frac{1200019893455926}{6241} a^{2} - \frac{1240009899654337}{6241} a - \frac{338045087886387}{6241} \) \( \bigl[a^{5} + a^{4} - 4 a^{3} - 4 a^{2} + 2 a + 3\) , \( a^{4} + a^{3} - 4 a^{2} - 2 a + 2\) , \( a^{2} + a - 2\) , \( -6 a^{5} + 13 a^{4} + 16 a^{3} - 44 a^{2} + 9 a + 11\) , \( 15 a^{5} - 33 a^{4} - 38 a^{3} + 107 a^{2} - 30 a - 14\bigr] \) ${y}^2+\left(a^{5}+a^{4}-4a^{3}-4a^{2}+2a+3\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{4}+a^{3}-4a^{2}-2a+2\right){x}^{2}+\left(-6a^{5}+13a^{4}+16a^{3}-44a^{2}+9a+11\right){x}+15a^{5}-33a^{4}-38a^{3}+107a^{2}-30a-14$
79.4-c2 79.4-c \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/10\Z$ $\mathrm{SU}(2)$ $1$ $91212.18964$ 1.49691 \( -\frac{9875329}{79} a^{5} - \frac{4876294}{79} a^{4} + 531385 a^{3} + \frac{23192531}{79} a^{2} - \frac{24136997}{79} a - \frac{6376947}{79} \) \( \bigl[a^{4} + a^{3} - 3 a^{2} - 2 a\) , \( -a^{3} - a^{2} + 3 a + 2\) , \( a^{5} - 4 a^{3} + a^{2} + 2 a - 2\) , \( a^{4} + a^{3} - 3 a^{2} - 4 a\) , \( -2 a^{5} + a^{4} + 12 a^{3} - a^{2} - 16 a - 5\bigr] \) ${y}^2+\left(a^{4}+a^{3}-3a^{2}-2a\right){x}{y}+\left(a^{5}-4a^{3}+a^{2}+2a-2\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+2\right){x}^{2}+\left(a^{4}+a^{3}-3a^{2}-4a\right){x}-2a^{5}+a^{4}+12a^{3}-a^{2}-16a-5$
79.4-c3 79.4-c \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $5.837580137$ 1.49691 \( \frac{111196987756273408746279}{3077056399} a^{5} - \frac{32180520780374974668601}{3077056399} a^{4} - \frac{7326933216778544198866}{38950081} a^{3} + \frac{33034429808135179499553}{3077056399} a^{2} + \frac{691511935344374409949404}{3077056399} a + \frac{158260508780641623871125}{3077056399} \) \( \bigl[a^{4} + a^{3} - 3 a^{2} - 2 a + 1\) , \( -a^{5} - a^{4} + 6 a^{3} + 4 a^{2} - 9 a - 1\) , \( a^{5} - 4 a^{3} + a^{2} + 3 a - 2\) , \( 8 a^{5} + 87 a^{4} - 136 a^{3} - 266 a^{2} + 327 a - 71\) , \( -465 a^{5} + 1504 a^{4} + 575 a^{3} - 4782 a^{2} + 2674 a + 6\bigr] \) ${y}^2+\left(a^{4}+a^{3}-3a^{2}-2a+1\right){x}{y}+\left(a^{5}-4a^{3}+a^{2}+3a-2\right){y}={x}^{3}+\left(-a^{5}-a^{4}+6a^{3}+4a^{2}-9a-1\right){x}^{2}+\left(8a^{5}+87a^{4}-136a^{3}-266a^{2}+327a-71\right){x}-465a^{5}+1504a^{4}+575a^{3}-4782a^{2}+2674a+6$
79.4-c4 79.4-c \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $2.918790068$ 1.49691 \( -\frac{59825344952154066993255580468150198478665}{9468276082626847201} a^{5} + \frac{165770769301487325035978806703005278360409}{9468276082626847201} a^{4} + \frac{70395843676331013626156886098889571019}{119851595982618319} a^{3} - \frac{249149902794951405348247878886512326245469}{9468276082626847201} a^{2} + \frac{82270495728478075345113362065223016794412}{9468276082626847201} a + \frac{33782222503946602790170526944985186954909}{9468276082626847201} \) \( \bigl[a^{4} + a^{3} - 3 a^{2} - 2 a + 1\) , \( -a^{5} - a^{4} + 6 a^{3} + 4 a^{2} - 9 a - 1\) , \( a^{5} - 4 a^{3} + a^{2} + 3 a - 2\) , \( -947 a^{5} + 2047 a^{4} + 2349 a^{3} - 6681 a^{2} + 2027 a + 739\) , \( -37043 a^{5} + 78539 a^{4} + 95061 a^{3} - 254944 a^{2} + 69625 a + 31719\bigr] \) ${y}^2+\left(a^{4}+a^{3}-3a^{2}-2a+1\right){x}{y}+\left(a^{5}-4a^{3}+a^{2}+3a-2\right){y}={x}^{3}+\left(-a^{5}-a^{4}+6a^{3}+4a^{2}-9a-1\right){x}^{2}+\left(-947a^{5}+2047a^{4}+2349a^{3}-6681a^{2}+2027a+739\right){x}-37043a^{5}+78539a^{4}+95061a^{3}-254944a^{2}+69625a+31719$
79.4-d1 79.4-d \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $12962.52150$ 1.32957 \( -\frac{1384680915160}{493039} a^{5} + \frac{1583778929801}{493039} a^{4} + \frac{81954702473}{6241} a^{3} - \frac{6526643131996}{493039} a^{2} - \frac{6414498868934}{493039} a + \frac{5372049895775}{493039} \) \( \bigl[a^{5} - 5 a^{3} + 5 a\) , \( -a^{5} + 4 a^{3} - 3 a - 1\) , \( a^{5} + a^{4} - 5 a^{3} - 4 a^{2} + 5 a + 2\) , \( -7 a^{5} + 3 a^{4} + 39 a^{3} - 4 a^{2} - 51 a - 13\) , \( 17 a^{5} - 5 a^{4} - 89 a^{3} + 5 a^{2} + 106 a + 23\bigr] \) ${y}^2+\left(a^{5}-5a^{3}+5a\right){x}{y}+\left(a^{5}+a^{4}-5a^{3}-4a^{2}+5a+2\right){y}={x}^{3}+\left(-a^{5}+4a^{3}-3a-1\right){x}^{2}+\left(-7a^{5}+3a^{4}+39a^{3}-4a^{2}-51a-13\right){x}+17a^{5}-5a^{4}-89a^{3}+5a^{2}+106a+23$
79.4-d2 79.4-d \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $25.31742482$ 1.32957 \( \frac{6020036548963366719669090043018}{59091511031674153381441} a^{5} - \frac{1511835207546230364924682979519}{59091511031674153381441} a^{4} - \frac{387616675912854011086810109059}{747993810527520928879} a^{3} + \frac{1833615268411384289237912829944}{59091511031674153381441} a^{2} + \frac{35817543189380927951633732407206}{59091511031674153381441} a + \frac{7335541199351009956876096226601}{59091511031674153381441} \) \( \bigl[a^{4} - 4 a^{2} + a + 2\) , \( a^{5} - 5 a^{3} + 4 a - 1\) , \( a^{5} - 5 a^{3} + 5 a + 1\) , \( 18 a^{5} - 77 a^{4} - 111 a^{3} + 338 a^{2} + 191 a - 262\) , \( 330 a^{5} - 411 a^{4} - 1588 a^{3} + 1727 a^{2} + 1683 a - 1469\bigr] \) ${y}^2+\left(a^{4}-4a^{2}+a+2\right){x}{y}+\left(a^{5}-5a^{3}+5a+1\right){y}={x}^{3}+\left(a^{5}-5a^{3}+4a-1\right){x}^{2}+\left(18a^{5}-77a^{4}-111a^{3}+338a^{2}+191a-262\right){x}+330a^{5}-411a^{4}-1588a^{3}+1727a^{2}+1683a-1469$
79.4-d3 79.4-d \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $12962.52150$ 1.32957 \( \frac{461179861176}{79} a^{5} - \frac{134120732904}{79} a^{4} - 30392416272 a^{3} + \frac{141924849311}{79} a^{2} + \frac{2867710847980}{79} a + \frac{650270453868}{79} \) \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( -a^{5} + 4 a^{3} - 3 a - 1\) , \( a^{5} + a^{4} - 4 a^{3} - 4 a^{2} + 2 a + 3\) , \( 7 a^{5} + 4 a^{4} - 32 a^{3} - 20 a^{2} + 22 a + 5\) , \( 18 a^{5} + 10 a^{4} - 78 a^{3} - 47 a^{2} + 47 a + 12\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-4a^{2}+2a+3\right){y}={x}^{3}+\left(-a^{5}+4a^{3}-3a-1\right){x}^{2}+\left(7a^{5}+4a^{4}-32a^{3}-20a^{2}+22a+5\right){x}+18a^{5}+10a^{4}-78a^{3}-47a^{2}+47a+12$
79.4-d4 79.4-d \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $1620.315188$ 1.32957 \( -\frac{60287849168080}{6241} a^{5} + \frac{1220116474110540}{6241} a^{4} - \frac{10979232465756}{79} a^{3} - \frac{2044220774210152}{6241} a^{2} + \frac{986299721072503}{6241} a + \frac{340251603176718}{6241} \) \( \bigl[a^{4} + a^{3} - 4 a^{2} - 3 a + 3\) , \( a^{5} - 5 a^{3} - a^{2} + 6 a + 3\) , \( a^{5} + a^{4} - 4 a^{3} - 4 a^{2} + 2 a + 2\) , \( -15 a^{5} + 13 a^{4} + 80 a^{3} - 40 a^{2} - 108 a - 19\) , \( 64 a^{5} + 2 a^{4} - 331 a^{3} - 62 a^{2} + 352 a + 85\bigr] \) ${y}^2+\left(a^{4}+a^{3}-4a^{2}-3a+3\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-4a^{2}+2a+2\right){y}={x}^{3}+\left(a^{5}-5a^{3}-a^{2}+6a+3\right){x}^{2}+\left(-15a^{5}+13a^{4}+80a^{3}-40a^{2}-108a-19\right){x}+64a^{5}+2a^{4}-331a^{3}-62a^{2}+352a+85$
79.4-d5 79.4-d \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $1620.315188$ 1.32957 \( -\frac{8438380784272797589233064}{243087455521} a^{5} + \frac{10472649598612580639218385}{243087455521} a^{4} + \frac{502116798593221367759733}{3077056399} a^{3} - \frac{43316234879703303465913356}{243087455521} a^{2} - \frac{40187894398552055242574886}{243087455521} a + \frac{35003373114582726388616286}{243087455521} \) \( \bigl[a^{3} - 2 a + 1\) , \( a^{5} - a^{4} - 5 a^{3} + 3 a^{2} + 6 a\) , \( a^{4} - 3 a^{2} + 1\) , \( -69 a^{5} + 3 a^{4} + 348 a^{3} + 13 a^{2} - 399 a - 94\) , \( -511 a^{5} + 93 a^{4} + 2528 a^{3} - 41 a^{2} - 2906 a - 664\bigr] \) ${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{4}-3a^{2}+1\right){y}={x}^{3}+\left(a^{5}-a^{4}-5a^{3}+3a^{2}+6a\right){x}^{2}+\left(-69a^{5}+3a^{4}+348a^{3}+13a^{2}-399a-94\right){x}-511a^{5}+93a^{4}+2528a^{3}-41a^{2}-2906a-664$
79.4-d6 79.4-d \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $25.31742482$ 1.32957 \( \frac{1100804881028314767283171320227}{38950081} a^{5} + \frac{1036830102461827114634407285921}{38950081} a^{4} - \frac{44185075908116339228656894532}{493039} a^{3} - \frac{2375160264558921129349900941591}{38950081} a^{2} + \frac{1992544437274547874569318216083}{38950081} a + \frac{566874791642885625266926884133}{38950081} \) \( \bigl[a^{3} - 3 a + 1\) , \( -a^{4} + 5 a^{2} + a - 4\) , \( a^{5} - 5 a^{3} + 6 a + 1\) , \( -2044 a^{5} + 604 a^{4} + 10657 a^{3} - 661 a^{2} - 12770 a - 2901\) , \( -10369 a^{5} + 629 a^{4} + 55366 a^{3} + 9036 a^{2} - 68550 a - 29027\bigr] \) ${y}^2+\left(a^{3}-3a+1\right){x}{y}+\left(a^{5}-5a^{3}+6a+1\right){y}={x}^{3}+\left(-a^{4}+5a^{2}+a-4\right){x}^{2}+\left(-2044a^{5}+604a^{4}+10657a^{3}-661a^{2}-12770a-2901\right){x}-10369a^{5}+629a^{4}+55366a^{3}+9036a^{2}-68550a-29027$
79.4-d7 79.4-d \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $202.5393985$ 1.32957 \( -\frac{7010369664218327367115661}{79} a^{5} + \frac{19425117724456157552636305}{79} a^{4} + 8249028023697825936892 a^{3} - \frac{29195534339945588288141479}{79} a^{2} + \frac{9640505769214320470363491}{79} a + \frac{3958620992765630783049989}{79} \) \( \bigl[a^{4} + a^{3} - 4 a^{2} - 3 a + 3\) , \( a^{5} - 5 a^{3} - a^{2} + 6 a + 3\) , \( a^{5} + a^{4} - 4 a^{3} - 4 a^{2} + 2 a + 2\) , \( 135 a^{5} + 58 a^{4} - 555 a^{3} - 310 a^{2} + 302 a + 71\) , \( 1650 a^{5} + 742 a^{4} - 7288 a^{3} - 3547 a^{2} + 4802 a + 1212\bigr] \) ${y}^2+\left(a^{4}+a^{3}-4a^{2}-3a+3\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-4a^{2}+2a+2\right){y}={x}^{3}+\left(a^{5}-5a^{3}-a^{2}+6a+3\right){x}^{2}+\left(135a^{5}+58a^{4}-555a^{3}-310a^{2}+302a+71\right){x}+1650a^{5}+742a^{4}-7288a^{3}-3547a^{2}+4802a+1212$
79.4-d8 79.4-d \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $202.5393985$ 1.32957 \( -\frac{2933202226852234845881866017482070}{493039} a^{5} + \frac{3640319144736813620991229392732945}{493039} a^{4} + \frac{174537052717825689312603206847501}{6241} a^{3} - \frac{15056831380161432327211321443712648}{493039} a^{2} - \frac{13969412421655742918255842956439418}{493039} a + \frac{12167259877404417312220612393046697}{493039} \) \( \bigl[a^{3} - 2 a + 1\) , \( a^{5} - a^{4} - 5 a^{3} + 3 a^{2} + 6 a\) , \( a^{4} - 3 a^{2} + 1\) , \( 31 a^{5} - 122 a^{4} - 102 a^{3} + 218 a^{2} + 156 a + 1\) , \( -1455 a^{5} + 1381 a^{4} + 6570 a^{3} - 2028 a^{2} - 7860 a - 1727\bigr] \) ${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{4}-3a^{2}+1\right){y}={x}^{3}+\left(a^{5}-a^{4}-5a^{3}+3a^{2}+6a\right){x}^{2}+\left(31a^{5}-122a^{4}-102a^{3}+218a^{2}+156a+1\right){x}-1455a^{5}+1381a^{4}+6570a^{3}-2028a^{2}-7860a-1727$
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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.