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Label Class Base field Conductor norm Rank Torsion CM Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
256.1-a1 256.1-a \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\mathsf{trivial}$ $1$ $1.915537937$ 4.059507167 \( -\frac{293180476215589246298781}{8} a - \frac{960141789432972248487021}{8} \) \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -10282063 a - 33672998\) , \( 34886965210 a + 114251923320\bigr] \) ${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-10282063a-33672998\right){x}+34886965210a+114251923320$
256.1-a2 256.1-a \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\mathsf{trivial}$ $1$ $1.915537937$ 4.059507167 \( \frac{293180476215589246298781}{8} a - \frac{626661132824280747392901}{4} \) \( \bigl[0\) , \( a + 1\) , \( 0\) , \( 10282065 a - 43955062\) , \( -34876683146 a + 149094933468\bigr] \) ${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(10282065a-43955062\right){x}-34876683146a+149094933468$
256.1-a3 256.1-a \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\mathsf{trivial}$ $1$ $1.915537937$ 4.059507167 \( -\frac{699691689}{2097152} a - \frac{307208349}{2097152} \) \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -3823 a - 12518\) , \( 369018 a + 1208504\bigr] \) ${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-3823a-12518\right){x}+369018a+1208504$
256.1-a4 256.1-a \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\mathsf{trivial}$ $1$ $1.915537937$ 4.059507167 \( \frac{699691689}{2097152} a - \frac{503450019}{1048576} \) \( \bigl[0\) , \( a + 1\) , \( 0\) , \( 3825 a - 16342\) , \( -365194 a + 1561180\bigr] \) ${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(3825a-16342\right){x}-365194a+1561180$
256.1-b1 256.1-b \(\Q(\sqrt{57}) \) \( 2^{8} \) $1$ $\Z/2\Z$ $2.349961086$ $2.170376093$ 2.702204615 \( -\frac{7754659}{1024} a + \frac{9383969}{512} \) \( \bigl[0\) , \( 1\) , \( 0\) , \( 44320 a - 189464\) , \( 9682784 a - 41393100\bigr] \) ${y}^2={x}^{3}+{x}^{2}+\left(44320a-189464\right){x}+9682784a-41393100$
256.1-b2 256.1-b \(\Q(\sqrt{57}) \) \( 2^{8} \) $1$ $\Z/2\Z$ $1.174980543$ $2.170376093$ 2.702204615 \( \frac{925430099}{32} a + \frac{1515353487}{16} \) \( \bigl[0\) , \( a - 1\) , \( 0\) , \( -3 a + 10\) , \( 204 a - 884\bigr] \) ${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-3a+10\right){x}+204a-884$
256.1-c1 256.1-c \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\Z/2\Z$ $1$ $0.789181095$ 1.881532613 \( -\frac{20297286875}{64} a + \frac{43383068421}{32} \) \( \bigl[0\) , \( a - 1\) , \( 0\) , \( 1461 a - 6246\) , \( 57028 a - 243796\bigr] \) ${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(1461a-6246\right){x}+57028a-243796$
256.1-c2 256.1-c \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\Z/2\Z$ $1$ $7.102629859$ 1.881532613 \( -\frac{489}{4} a + 1841 \) \( \bigl[0\) , \( a - 1\) , \( 0\) , \( 21 a - 86\) , \( 20 a - 84\bigr] \) ${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(21a-86\right){x}+20a-84$
256.1-c3 256.1-c \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\Z/2\Z$ $1$ $7.102629859$ 1.881532613 \( \frac{489}{4} a + \frac{6875}{4} \) \( \bigl[0\) , \( -a\) , \( 0\) , \( -21 a - 65\) , \( -20 a - 64\bigr] \) ${y}^2={x}^{3}-a{x}^{2}+\left(-21a-65\right){x}-20a-64$
256.1-c4 256.1-c \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\Z/2\Z$ $1$ $0.789181095$ 1.881532613 \( \frac{20297286875}{64} a + \frac{66468849967}{64} \) \( \bigl[0\) , \( -a\) , \( 0\) , \( -1461 a - 4785\) , \( -57028 a - 186768\bigr] \) ${y}^2={x}^{3}-a{x}^{2}+\left(-1461a-4785\right){x}-57028a-186768$
256.1-d1 256.1-d \(\Q(\sqrt{57}) \) \( 2^{8} \) $1$ $\mathsf{trivial}$ $0.270502588$ $3.783765348$ 2.169093018 \( -\frac{4171}{2} a - 5149 \) \( \bigl[0\) , \( a\) , \( 0\) , \( -21 a - 65\) , \( -140 a - 460\bigr] \) ${y}^2={x}^{3}+a{x}^{2}+\left(-21a-65\right){x}-140a-460$
256.1-e1 256.1-e \(\Q(\sqrt{57}) \) \( 2^{8} \) $1$ $\mathsf{trivial}$ $0.523959333$ $20.62619196$ 2.862919944 \( 7168 a - 30720 \) \( \bigl[0\) , \( -a\) , \( 0\) , \( 2823 a - 12062\) , \( -158029 a + 675558\bigr] \) ${y}^2={x}^{3}-a{x}^{2}+\left(2823a-12062\right){x}-158029a+675558$
256.1-f1 256.1-f \(\Q(\sqrt{57}) \) \( 2^{8} \) $1$ $\mathsf{trivial}$ $0.270502588$ $3.783765348$ 2.169093018 \( \frac{4171}{2} a - \frac{14469}{2} \) \( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 21 a - 86\) , \( 140 a - 600\bigr] \) ${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(21a-86\right){x}+140a-600$
256.1-g1 256.1-g \(\Q(\sqrt{57}) \) \( 2^{8} \) $1$ $\mathsf{trivial}$ $0.523959333$ $20.62619196$ 2.862919944 \( -7168 a - 23552 \) \( \bigl[0\) , \( a - 1\) , \( 0\) , \( -2823 a - 9239\) , \( 158029 a + 517529\bigr] \) ${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-2823a-9239\right){x}+158029a+517529$
256.1-h1 256.1-h \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\Z/2\Z$ $-3$ $1$ $17.69503190$ 0.585941058 \( 0 \) \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a + 5\) , \( 133 a + 436\bigr] \) ${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+5\right){x}+133a+436$
256.1-h2 256.1-h \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\Z/2\Z$ $-3$ $1$ $17.69503190$ 0.585941058 \( 0 \) \( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 5\) , \( -133 a + 574\bigr] \) ${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a+5\right){x}-133a+574$
256.1-h3 256.1-h \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\Z/2\Z$ $-12$ $1$ $17.69503190$ 0.585941058 \( 54000 \) \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -199 a - 650\) , \( 3294 a + 10788\bigr] \) ${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-199a-650\right){x}+3294a+10788$
256.1-h4 256.1-h \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\Z/2\Z$ $-12$ $1$ $17.69503190$ 0.585941058 \( 54000 \) \( \bigl[0\) , \( a + 1\) , \( 0\) , \( 201 a - 850\) , \( -3094 a + 13232\bigr] \) ${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(201a-850\right){x}-3094a+13232$
256.1-i1 256.1-i \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\mathsf{trivial}$ $-3$ $1$ $8.847515954$ 2.343764232 \( 0 \) \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a + 5\) , \( 93 a - 403\bigr] \) ${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+5\right){x}+93a-403$
256.1-i2 256.1-i \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\mathsf{trivial}$ $-3$ $1$ $8.847515954$ 2.343764232 \( 0 \) \( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 5\) , \( -12213 a - 39997\bigr] \) ${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a+5\right){x}-12213a-39997$
256.1-j1 256.1-j \(\Q(\sqrt{57}) \) \( 2^{8} \) $1$ $\Z/2\Z$ $1.174980543$ $2.170376093$ 2.702204615 \( -\frac{925430099}{32} a + \frac{3956137073}{32} \) \( \bigl[0\) , \( -a\) , \( 0\) , \( 3 a + 7\) , \( -204 a - 680\bigr] \) ${y}^2={x}^{3}-a{x}^{2}+\left(3a+7\right){x}-204a-680$
256.1-j2 256.1-j \(\Q(\sqrt{57}) \) \( 2^{8} \) $1$ $\Z/2\Z$ $2.349961086$ $2.170376093$ 2.702204615 \( \frac{7754659}{1024} a + \frac{11013279}{1024} \) \( \bigl[0\) , \( 1\) , \( 0\) , \( -44320 a - 145144\) , \( -9682784 a - 31710316\bigr] \) ${y}^2={x}^{3}+{x}^{2}+\left(-44320a-145144\right){x}-9682784a-31710316$
256.1-k1 256.1-k \(\Q(\sqrt{57}) \) \( 2^{8} \) $1$ $\mathsf{trivial}$ $0.156656317$ $11.06868076$ 3.674737569 \( \frac{4171}{2} a - \frac{14469}{2} \) \( \bigl[0\) , \( -a\) , \( 0\) , \( 2395 a + 7847\) , \( -24172 a - 79160\bigr] \) ${y}^2={x}^{3}-a{x}^{2}+\left(2395a+7847\right){x}-24172a-79160$
256.1-l1 256.1-l \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\mathsf{trivial}$ $-3$ $1$ $8.847515954$ 2.343764232 \( 0 \) \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a + 5\) , \( 12213 a - 52215\bigr] \) ${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+5\right){x}+12213a-52215$
256.1-l2 256.1-l \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\mathsf{trivial}$ $-3$ $1$ $8.847515954$ 2.343764232 \( 0 \) \( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 5\) , \( -93 a - 305\bigr] \) ${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a+5\right){x}-93a-305$
256.1-m1 256.1-m \(\Q(\sqrt{57}) \) \( 2^{8} \) $1$ $\Z/2\Z$ $0.979895166$ $11.24713795$ 5.839076984 \( -\frac{925430099}{32} a + \frac{3956137073}{32} \) \( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 77309 a - 330486\) , \( -22623164 a + 96712152\bigr] \) ${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(77309a-330486\right){x}-22623164a+96712152$
256.1-m2 256.1-m \(\Q(\sqrt{57}) \) \( 2^{8} \) $1$ $\Z/2\Z$ $0.489947583$ $11.24713795$ 5.839076984 \( \frac{7754659}{1024} a + \frac{11013279}{1024} \) \( \bigl[0\) , \( a\) , \( 0\) , \( 27 a - 113\) , \( -188 a + 804\bigr] \) ${y}^2={x}^{3}+a{x}^{2}+\left(27a-113\right){x}-188a+804$
256.1-n1 256.1-n \(\Q(\sqrt{57}) \) \( 2^{8} \) $1$ $\mathsf{trivial}$ $7.564554459$ $2.988785561$ 5.989225681 \( -7168 a - 23552 \) \( \bigl[0\) , \( -1\) , \( 0\) , \( 4 a - 17\) , \( 40 a - 171\bigr] \) ${y}^2={x}^{3}-{x}^{2}+\left(4a-17\right){x}+40a-171$
256.1-o1 256.1-o \(\Q(\sqrt{57}) \) \( 2^{8} \) $1$ $\mathsf{trivial}$ $7.564554459$ $2.988785561$ 5.989225681 \( 7168 a - 30720 \) \( \bigl[0\) , \( -1\) , \( 0\) , \( -4 a - 13\) , \( -40 a - 131\bigr] \) ${y}^2={x}^{3}-{x}^{2}+\left(-4a-13\right){x}-40a-131$
256.1-p1 256.1-p \(\Q(\sqrt{57}) \) \( 2^{8} \) $1$ $\mathsf{trivial}$ $0.156656317$ $11.06868076$ 3.674737569 \( -\frac{4171}{2} a - 5149 \) \( \bigl[0\) , \( a - 1\) , \( 0\) , \( -2395 a + 10242\) , \( 24172 a - 103332\bigr] \) ${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-2395a+10242\right){x}+24172a-103332$
256.1-q1 256.1-q \(\Q(\sqrt{57}) \) \( 2^{8} \) $1$ $\Z/2\Z$ $0.489947583$ $11.24713795$ 5.839076984 \( -\frac{7754659}{1024} a + \frac{9383969}{512} \) \( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -27 a - 86\) , \( 188 a + 616\bigr] \) ${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-27a-86\right){x}+188a+616$
256.1-q2 256.1-q \(\Q(\sqrt{57}) \) \( 2^{8} \) $1$ $\Z/2\Z$ $0.979895166$ $11.24713795$ 5.839076984 \( \frac{925430099}{32} a + \frac{1515353487}{16} \) \( \bigl[0\) , \( a\) , \( 0\) , \( -77309 a - 253177\) , \( 22623164 a + 74088988\bigr] \) ${y}^2={x}^{3}+a{x}^{2}+\left(-77309a-253177\right){x}+22623164a+74088988$
256.1-r1 256.1-r \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\mathsf{trivial}$ $1$ $0.019466671$ 0.505371038 \( -\frac{293180476215589246298781}{8} a - \frac{960141789432972248487021}{8} \) \( \bigl[0\) , \( a + 1\) , \( 0\) , \( 1142337 a - 4884238\) , \( 1290582430 a - 5517144100\bigr] \) ${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(1142337a-4884238\right){x}+1290582430a-5517144100$
256.1-r2 256.1-r \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\mathsf{trivial}$ $1$ $0.019466671$ 0.505371038 \( \frac{293180476215589246298781}{8} a - \frac{626661132824280747392901}{4} \) \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -1142335 a - 3741902\) , \( -1289440094 a - 4222819768\bigr] \) ${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1142335a-3741902\right){x}-1289440094a-4222819768$
256.1-r3 256.1-r \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\mathsf{trivial}$ $1$ $0.953866916$ 0.505371038 \( -\frac{699691689}{2097152} a - \frac{307208349}{2097152} \) \( \bigl[0\) , \( a + 1\) , \( 0\) , \( -3423 a + 14642\) , \( -190018 a + 812316\bigr] \) ${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-3423a+14642\right){x}-190018a+812316$
256.1-r4 256.1-r \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\mathsf{trivial}$ $1$ $0.953866916$ 0.505371038 \( \frac{699691689}{2097152} a - \frac{503450019}{1048576} \) \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 3425 a + 11218\) , \( 186594 a + 611080\bigr] \) ${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(3425a+11218\right){x}+186594a+611080$
256.1-s1 256.1-s \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\Z/2\Z$ $1$ $9.323493311$ 4.939707428 \( -\frac{20297286875}{64} a + \frac{43383068421}{32} \) \( \bigl[0\) , \( a\) , \( 0\) , \( -47365 a - 155113\) , \( 11028284 a + 36116716\bigr] \) ${y}^2={x}^{3}+a{x}^{2}+\left(-47365a-155113\right){x}+11028284a+36116716$
256.1-s2 256.1-s \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\Z/2\Z$ $1$ $9.323493311$ 4.939707428 \( -\frac{489}{4} a + 1841 \) \( \bigl[0\) , \( a\) , \( 0\) , \( 2395 a + 7847\) , \( 72492 a + 237404\bigr] \) ${y}^2={x}^{3}+a{x}^{2}+\left(2395a+7847\right){x}+72492a+237404$
256.1-s3 256.1-s \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\Z/2\Z$ $1$ $9.323493311$ 4.939707428 \( \frac{489}{4} a + \frac{6875}{4} \) \( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -2395 a + 10242\) , \( -72492 a + 309896\bigr] \) ${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2395a+10242\right){x}-72492a+309896$
256.1-s4 256.1-s \(\Q(\sqrt{57}) \) \( 2^{8} \) $0$ $\Z/2\Z$ $1$ $9.323493311$ 4.939707428 \( \frac{20297286875}{64} a + \frac{66468849967}{64} \) \( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 47365 a - 202478\) , \( -11028284 a + 47145000\bigr] \) ${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(47365a-202478\right){x}-11028284a+47145000$
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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.