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Results (6 matches)

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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
9.2-a1 9.2-a \(\Q(\sqrt{19}) \) \( 3^{2} \) 0 $\Z/2\Z$ $-4$ $N(\mathrm{U}(1))$ $1$ $29.55147182$ 1.694893148 \( 1728 \) \( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 2 a + 32\) , \( 9 a + 29\bigr] \) ${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2a+32\right){x}+9a+29$
9.2-a2 9.2-a \(\Q(\sqrt{19}) \) \( 3^{2} \) 0 $\Z/2\Z$ $-4$ $N(\mathrm{U}(1))$ $1$ $29.55147182$ 1.694893148 \( 1728 \) \( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 87 a + 381\) , \( 349 a + 1521\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(87a+381\right){x}+349a+1521$
9.2-b1 9.2-b \(\Q(\sqrt{19}) \) \( 3^{2} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1.105793423$ $6.013673307$ 3.051174382 \( -\frac{53248}{81} a - \frac{131072}{81} \) \( \bigl[0\) , \( 0\) , \( a\) , \( -4759 a + 20744\) , \( 231623 a - 1009626\bigr] \) ${y}^2+a{y}={x}^{3}+\left(-4759a+20744\right){x}+231623a-1009626$
9.2-c1 9.2-c \(\Q(\sqrt{19}) \) \( 3^{2} \) $1$ $\mathsf{trivial}$ $-19$ $N(\mathrm{U}(1))$ $1.624922199$ $2.325972790$ 1.734164921 \( -884736 \) \( \bigl[0\) , \( 0\) , \( a\) , \( -16 a - 70\) , \( -73 a - 323\bigr] \) ${y}^2+a{y}={x}^{3}+\left(-16a-70\right){x}-73a-323$
9.2-c2 9.2-c \(\Q(\sqrt{19}) \) \( 3^{2} \) $1$ $\mathsf{trivial}$ $-19$ $N(\mathrm{U}(1))$ $0.085522221$ $44.19348301$ 1.734164921 \( -884736 \) \( \bigl[0\) , \( 0\) , \( 1\) , \( -16 a - 70\) , \( 73 a + 318\bigr] \) ${y}^2+{y}={x}^{3}+\left(-16a-70\right){x}+73a+318$
9.2-d1 9.2-d \(\Q(\sqrt{19}) \) \( 3^{2} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.098423053$ $19.00729698$ 0.858361815 \( -\frac{53248}{81} a - \frac{131072}{81} \) \( \bigl[0\) , \( 0\) , \( 1\) , \( -4759 a + 20744\) , \( -231623 a + 1009621\bigr] \) ${y}^2+{y}={x}^{3}+\left(-4759a+20744\right){x}-231623a+1009621$
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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.