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Results (18 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
441.1-a1 441.1-a \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 7^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $0.819338121$ 0.357588472 \( -\frac{1381357}{441} a - \frac{51890}{9} \) \( \bigl[1\) , \( a\) , \( 1\) , \( 151 a - 421\) , \( 5254 a - 14668\bigr] \) ${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(151a-421\right){x}+5254a-14668$
441.1-a2 441.1-a \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 7^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $0.819338121$ 0.357588472 \( \frac{28686222041}{567} a + \frac{5735365645}{63} \) \( \bigl[1\) , \( a\) , \( 1\) , \( 3826 a - 10711\) , \( 193120 a - 539164\bigr] \) ${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(3826a-10711\right){x}+193120a-539164$
441.1-b1 441.1-b \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 7^{2} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $2.370416734$ $0.481459439$ 2.988518917 \( -\frac{1713910976512}{1594323} \) \( \bigl[0\) , \( -a - 1\) , \( 1\) , \( -2736 a - 5472\) , \( 131031 a + 229304\bigr] \) ${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2736a-5472\right){x}+131031a+229304$
441.1-b2 441.1-b \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 7^{2} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.182339748$ $6.258972717$ 2.988518917 \( -\frac{28672}{3} \) \( \bigl[0\) , \( -a - 1\) , \( 1\) , \( -6 a - 12\) , \( -9 a - 16\bigr] \) ${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6a-12\right){x}-9a-16$
441.1-c1 441.1-c \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 7^{2} \) 0 $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $1$ $5.709422123$ 2.491796100 \( -3375 \) \( \bigl[a\) , \( -a + 1\) , \( a\) , \( -8 a - 13\) , \( -19 a - 34\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-8a-13\right){x}-19a-34$
441.1-c2 441.1-c \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 7^{2} \) 0 $\Z/2\Z$ $-7$ $N(\mathrm{U}(1))$ $1$ $5.709422123$ 2.491796100 \( -3375 \) \( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 6 a - 21\) , \( 18 a - 53\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(6a-21\right){x}+18a-53$
441.1-c3 441.1-c \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 7^{2} \) 0 $\Z/2\Z$ $-28$ $N(\mathrm{U}(1))$ $1$ $5.709422123$ 2.491796100 \( 16581375 \) \( \bigl[a\) , \( -a + 1\) , \( a\) , \( -113 a - 223\) , \( -1048 a - 1861\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-113a-223\right){x}-1048a-1861$
441.1-c4 441.1-c \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 7^{2} \) 0 $\Z/2\Z$ $-28$ $N(\mathrm{U}(1))$ $1$ $5.709422123$ 2.491796100 \( 16581375 \) \( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 111 a - 336\) , \( 1047 a - 2909\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(111a-336\right){x}+1047a-2909$
441.1-d1 441.1-d \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 7^{2} \) $1$ $\mathsf{trivial}$ $-3$ $N(\mathrm{U}(1))$ $0.298091926$ $2.558692321$ 1.997284256 \( 0 \) \( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( -294 a - 527\bigr] \) ${y}^2+{y}={x}^{3}-294a-527$
441.1-d2 441.1-d \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 7^{2} \) $1$ $\Z/3\Z$ $-3$ $N(\mathrm{U}(1))$ $0.894275780$ $7.676076964$ 1.997284256 \( 0 \) \( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 12\bigr] \) ${y}^2+{y}={x}^{3}+12$
441.1-e1 441.1-e \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 7^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $5.167093490$ 2.255104480 \( \frac{1381357}{441} a - \frac{1307989}{147} \) \( \bigl[a\) , \( -1\) , \( 0\) , \( 4 a - 41\) , \( 16 a + 168\bigr] \) ${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(4a-41\right){x}+16a+168$
441.1-e2 441.1-e \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 7^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $5.167093490$ 2.255104480 \( -\frac{28686222041}{567} a + \frac{80304512846}{567} \) \( \bigl[a\) , \( -1\) , \( 0\) , \( 4 a - 776\) , \( -131 a + 8253\bigr] \) ${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(4a-776\right){x}-131a+8253$
441.1-f1 441.1-f \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 7^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $5.167093490$ 2.255104480 \( -\frac{1381357}{441} a - \frac{51890}{9} \) \( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -4 a - 37\) , \( -16 a + 184\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-4a-37\right){x}-16a+184$
441.1-f2 441.1-f \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 7^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $5.167093490$ 2.255104480 \( \frac{28686222041}{567} a + \frac{5735365645}{63} \) \( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -4 a - 772\) , \( 131 a + 8122\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-4a-772\right){x}+131a+8122$
441.1-g1 441.1-g \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 7^{2} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $2.370416734$ $0.481459439$ 2.988518917 \( -\frac{1713910976512}{1594323} \) \( \bigl[0\) , \( a + 1\) , \( 1\) , \( 2738 a - 8209\) , \( -128294 a + 352126\bigr] \) ${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2738a-8209\right){x}-128294a+352126$
441.1-g2 441.1-g \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 7^{2} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.182339748$ $6.258972717$ 2.988518917 \( -\frac{28672}{3} \) \( \bigl[0\) , \( a + 1\) , \( 1\) , \( 8 a - 19\) , \( 16 a - 44\bigr] \) ${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(8a-19\right){x}+16a-44$
441.1-h1 441.1-h \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 7^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $0.819338121$ 0.357588472 \( \frac{1381357}{441} a - \frac{1307989}{147} \) \( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -151 a - 270\) , \( -5254 a - 9414\bigr] \) ${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-151a-270\right){x}-5254a-9414$
441.1-h2 441.1-h \(\Q(\sqrt{21}) \) \( 3^{2} \cdot 7^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $0.819338121$ 0.357588472 \( -\frac{28686222041}{567} a + \frac{80304512846}{567} \) \( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -3826 a - 6885\) , \( -193120 a - 346044\bigr] \) ${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-3826a-6885\right){x}-193120a-346044$
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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.