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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
57.1-a1 57.1-a 4.4.19821.1 \( 3 \cdot 19 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.026666755$ $2988.763883$ 4.528860944 \( \frac{764816735}{19} a^{3} - \frac{2749538048}{19} a^{2} + \frac{4046607218}{57} a + \frac{2502325034}{57} \) \( \bigl[-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 3\) , \( -\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 4 a\) , \( a^{2} - a - 3\) , \( -\frac{1}{3} a^{3} - \frac{7}{3} a^{2} - 2 a - 4\) , \( 2 a^{3} + 5 a^{2} - 3 a - 3\bigr] \) ${y}^2+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+2a-3\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+4a\right){x}^{2}+\left(-\frac{1}{3}a^{3}-\frac{7}{3}a^{2}-2a-4\right){x}+2a^{3}+5a^{2}-3a-3$
57.1-b1 57.1-b 4.4.19821.1 \( 3 \cdot 19 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.254490339$ $181.8351186$ 2.629518748 \( -\frac{43021}{57} a^{3} + \frac{85696}{57} a^{2} + \frac{369034}{57} a - \frac{454327}{57} \) \( \bigl[a + 1\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a - 1\) , \( a\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\) , \( 2 a - 3\bigr] \) ${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-3a-1\right){x}^{2}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-3\right){x}+2a-3$
57.1-c1 57.1-c 4.4.19821.1 \( 3 \cdot 19 \) $0 \le r \le 1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $18.86826300$ 3.895783893 \( -\frac{404963364612649}{4617} a^{3} + \frac{4913008494921869}{41553} a^{2} + \frac{27447645111084058}{41553} a - \frac{31419748638630586}{41553} \) \( \bigl[a^{2} - a - 4\) , \( -\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a\) , \( a^{2} - a - 4\) , \( \frac{106}{3} a^{3} + \frac{10}{3} a^{2} - 278 a - 102\) , \( 883 a^{3} + 93 a^{2} - 6961 a - 2401\bigr] \) ${y}^2+\left(a^{2}-a-4\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+3a\right){x}^{2}+\left(\frac{106}{3}a^{3}+\frac{10}{3}a^{2}-278a-102\right){x}+883a^{3}+93a^{2}-6961a-2401$
57.1-d1 57.1-d 4.4.19821.1 \( 3 \cdot 19 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $1.949888097$ $90.41503854$ 5.008960080 \( \frac{4957694999417395}{46619598843} a^{3} + \frac{9593307021539665}{46619598843} a^{2} - \frac{11878525618289761}{46619598843} a - \frac{5175805466469244}{46619598843} \) \( \bigl[-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 3\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a - 1\) , \( 305 a^{3} + 31 a^{2} - 2403 a - 828\) , \( -3302 a^{3} - 350 a^{2} + 26036 a + 8964\bigr] \) ${y}^2+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+2a-3\right){x}{y}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a-1\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-3\right){x}^{2}+\left(305a^{3}+31a^{2}-2403a-828\right){x}-3302a^{3}-350a^{2}+26036a+8964$
57.1-d2 57.1-d 4.4.19821.1 \( 3 \cdot 19 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $3.899776194$ $180.8300770$ 5.008960080 \( -\frac{217217028070}{373977} a^{3} + \frac{1324842416596}{373977} a^{2} - \frac{2515509051217}{373977} a + \frac{161738260330}{41553} \) \( \bigl[-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 3\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a - 1\) , \( -\frac{190}{3} a^{3} - \frac{22}{3} a^{2} + 502 a + 172\) , \( -\frac{1208}{3} a^{3} - \frac{134}{3} a^{2} + 3179 a + 1095\bigr] \) ${y}^2+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+2a-3\right){x}{y}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a-1\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+3a-3\right){x}^{2}+\left(-\frac{190}{3}a^{3}-\frac{22}{3}a^{2}+502a+172\right){x}-\frac{1208}{3}a^{3}-\frac{134}{3}a^{2}+3179a+1095$
57.1-e1 57.1-e 4.4.19821.1 \( 3 \cdot 19 \) $2$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.199016654$ $115.2853003$ 5.214954858 \( \frac{732466}{171} a^{3} + \frac{3616163}{513} a^{2} - \frac{6735095}{513} a - \frac{2042806}{513} \) \( \bigl[\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a\) , \( a^{2} - 2 a - 5\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a - 1\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 6\) , \( \frac{1}{3} a^{3} + \frac{4}{3} a^{2} - 4 a - 5\bigr] \) ${y}^2+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-2a\right){x}{y}+\left(\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-3a-1\right){y}={x}^{3}+\left(a^{2}-2a-5\right){x}^{2}+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+6\right){x}+\frac{1}{3}a^{3}+\frac{4}{3}a^{2}-4a-5$
57.1-f1 57.1-f 4.4.19821.1 \( 3 \cdot 19 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.241538489$ $306.1728257$ 4.202233631 \( \frac{14227698499096}{57} a^{3} + \frac{9089735445467}{19} a^{2} - \frac{34285899745478}{57} a - \frac{14634193939504}{57} \) \( \bigl[-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 3\) , \( \frac{2}{3} a^{3} - \frac{1}{3} a^{2} - 6 a + 3\) , \( a^{2} - a - 3\) , \( 2 a^{3} - 5 a^{2} - 13 a + 16\) , \( \frac{4}{3} a^{3} - \frac{14}{3} a^{2} - 12 a + 22\bigr] \) ${y}^2+\left(-\frac{1}{3}a^{3}+\frac{2}{3}a^{2}+2a-3\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(\frac{2}{3}a^{3}-\frac{1}{3}a^{2}-6a+3\right){x}^{2}+\left(2a^{3}-5a^{2}-13a+16\right){x}+\frac{4}{3}a^{3}-\frac{14}{3}a^{2}-12a+22$
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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.