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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
45.4-a1 45.4-a 4.4.4525.1 \( 3^{2} \cdot 5 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.451118733$ $24.99201932$ 2.681654982 \( -\frac{61709015057278}{1875} a^{3} + \frac{568055547171271}{5625} a^{2} + \frac{40298462678968}{1875} a - \frac{17899615289469}{125} \) \( \bigl[\frac{1}{3} a^{3} + \frac{2}{3} a^{2} - \frac{4}{3} a - 4\) , \( \frac{1}{3} a^{3} + \frac{2}{3} a^{2} - \frac{4}{3} a - 4\) , \( a + 1\) , \( 7 a^{3} + a^{2} - 53 a - 53\) , \( 68 a^{3} + 20 a^{2} - 466 a - 448\bigr] \) ${y}^2+\left(\frac{1}{3}a^{3}+\frac{2}{3}a^{2}-\frac{4}{3}a-4\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(\frac{1}{3}a^{3}+\frac{2}{3}a^{2}-\frac{4}{3}a-4\right){x}^{2}+\left(7a^{3}+a^{2}-53a-53\right){x}+68a^{3}+20a^{2}-466a-448$
45.4-a2 45.4-a 4.4.4525.1 \( 3^{2} \cdot 5 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.902237467$ $49.98403864$ 2.681654982 \( -\frac{1262963}{25} a^{3} - \frac{3615727}{25} a^{2} + \frac{51598387}{75} a + \frac{21590004}{25} \) \( \bigl[1\) , \( \frac{1}{3} a^{3} + \frac{2}{3} a^{2} - \frac{4}{3} a - 4\) , \( 1\) , \( -\frac{65}{3} a^{3} + \frac{155}{3} a^{2} + \frac{296}{3} a - 209\) , \( -156 a^{3} + 334 a^{2} + 727 a - 1306\bigr] \) ${y}^2+{x}{y}+{y}={x}^{3}+\left(\frac{1}{3}a^{3}+\frac{2}{3}a^{2}-\frac{4}{3}a-4\right){x}^{2}+\left(-\frac{65}{3}a^{3}+\frac{155}{3}a^{2}+\frac{296}{3}a-209\right){x}-156a^{3}+334a^{2}+727a-1306$
45.4-b1 45.4-b 4.4.4525.1 \( 3^{2} \cdot 5 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.015271155$ $472.5674111$ 2.145637649 \( \frac{311063}{405} a^{3} + \frac{426403}{243} a^{2} - \frac{2175776}{405} a - \frac{1637564}{135} \) \( \bigl[a^{2} - a - 4\) , \( -a^{2} + a + 5\) , \( \frac{1}{3} a^{3} + \frac{2}{3} a^{2} - \frac{4}{3} a - 3\) , \( -\frac{44}{3} a^{3} + \frac{86}{3} a^{2} + \frac{209}{3} a - 108\) , \( 56 a^{3} - 120 a^{2} - 262 a + 467\bigr] \) ${y}^2+\left(a^{2}-a-4\right){x}{y}+\left(\frac{1}{3}a^{3}+\frac{2}{3}a^{2}-\frac{4}{3}a-3\right){y}={x}^{3}+\left(-a^{2}+a+5\right){x}^{2}+\left(-\frac{44}{3}a^{3}+\frac{86}{3}a^{2}+\frac{209}{3}a-108\right){x}+56a^{3}-120a^{2}-262a+467$
45.4-b2 45.4-b 4.4.4525.1 \( 3^{2} \cdot 5 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.076355779$ $472.5674111$ 2.145637649 \( -\frac{23573637}{125} a^{3} - \frac{24357233}{75} a^{2} + \frac{54667564}{125} a + \frac{78032973}{125} \) \( \bigl[1\) , \( a^{2} - a - 4\) , \( \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{4}{3} a + 1\) , \( -a^{3} + 5 a + 4\) , \( \frac{20}{3} a^{3} + \frac{10}{3} a^{2} - \frac{128}{3} a - 43\bigr] \) ${y}^2+{x}{y}+\left(\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{4}{3}a+1\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-a^{3}+5a+4\right){x}+\frac{20}{3}a^{3}+\frac{10}{3}a^{2}-\frac{128}{3}a-43$
45.4-c1 45.4-c 4.4.4525.1 \( 3^{2} \cdot 5 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.007009552$ $1579.532167$ 1.975107543 \( \frac{95132237}{225} a^{3} - \frac{217836256}{135} a^{2} + \frac{232697396}{225} a + \frac{45330989}{75} \) \( \bigl[\frac{1}{3} a^{3} + \frac{2}{3} a^{2} - \frac{7}{3} a - 4\) , \( -\frac{1}{3} a^{3} + \frac{4}{3} a^{2} + \frac{4}{3} a - 3\) , \( a^{2} - 4\) , \( -\frac{4}{3} a^{3} + \frac{1}{3} a^{2} + \frac{25}{3} a + 4\) , \( -\frac{5}{3} a^{3} - \frac{4}{3} a^{2} + \frac{32}{3} a + 13\bigr] \) ${y}^2+\left(\frac{1}{3}a^{3}+\frac{2}{3}a^{2}-\frac{7}{3}a-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}+\frac{4}{3}a^{2}+\frac{4}{3}a-3\right){x}^{2}+\left(-\frac{4}{3}a^{3}+\frac{1}{3}a^{2}+\frac{25}{3}a+4\right){x}-\frac{5}{3}a^{3}-\frac{4}{3}a^{2}+\frac{32}{3}a+13$
45.4-c2 45.4-c 4.4.4525.1 \( 3^{2} \cdot 5 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.021028658$ $1579.532167$ 1.975107543 \( -\frac{1037039198606}{15} a^{3} - 118962068606 a^{2} + \frac{2404382003012}{15} a + \frac{1143498460323}{5} \) \( \bigl[a^{2} - 3\) , \( \frac{1}{3} a^{3} - \frac{4}{3} a^{2} - \frac{1}{3} a + 3\) , \( a\) , \( \frac{10}{3} a^{3} + \frac{8}{3} a^{2} - \frac{73}{3} a - 24\) , \( -8 a^{3} - 6 a^{2} + 53 a + 60\bigr] \) ${y}^2+\left(a^{2}-3\right){x}{y}+a{y}={x}^{3}+\left(\frac{1}{3}a^{3}-\frac{4}{3}a^{2}-\frac{1}{3}a+3\right){x}^{2}+\left(\frac{10}{3}a^{3}+\frac{8}{3}a^{2}-\frac{73}{3}a-24\right){x}-8a^{3}-6a^{2}+53a+60$
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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.