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Results (28 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
448.1-a1 448.1-a \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $11.37856052$ 2.483005471 \( -\frac{4}{7} \) \( \bigl[0\) , \( -a\) , \( 0\) , \( 2 a - 3\) , \( -97 a + 270\bigr] \) ${y}^2={x}^{3}-a{x}^{2}+\left(2a-3\right){x}-97a+270$
448.1-a2 448.1-a \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $11.37856052$ 2.483005471 \( \frac{3543122}{49} \) \( \bigl[0\) , \( -a\) , \( 0\) , \( 202 a - 563\) , \( -2217 a + 6190\bigr] \) ${y}^2={x}^{3}-a{x}^{2}+\left(202a-563\right){x}-2217a+6190$
448.1-b1 448.1-b \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) $1$ $\Z/4\Z$ $\mathrm{SU}(2)$ $1.530313332$ $8.919127558$ 2.978469037 \( -\frac{55296}{49} \) \( \bigl[0\) , \( 0\) , \( 0\) , \( 2 a - 6\) , \( 4 a - 11\bigr] \) ${y}^2={x}^{3}+\left(2a-6\right){x}+4a-11$
448.1-b2 448.1-b \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $6.121253330$ $2.229781889$ 2.978469037 \( -\frac{14919813068184}{7} a + \frac{41645492922276}{7} \) \( \bigl[0\) , \( 0\) , \( 0\) , \( 617 a - 1731\) , \( 12688 a - 35410\bigr] \) ${y}^2={x}^{3}+\left(617a-1731\right){x}+12688a-35410$
448.1-b3 448.1-b \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) $1$ $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $3.060626665$ $8.919127558$ 2.978469037 \( \frac{21882096}{7} \) \( \bigl[0\) , \( 0\) , \( 0\) , \( 37 a - 111\) , \( 200 a - 550\bigr] \) ${y}^2={x}^{3}+\left(37a-111\right){x}+200a-550$
448.1-b4 448.1-b \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $1.530313332$ $8.919127558$ 2.978469037 \( \frac{14919813068184}{7} a + \frac{26725679854092}{7} \) \( \bigl[0\) , \( 0\) , \( 0\) , \( 17 a - 171\) , \( 256 a - 186\bigr] \) ${y}^2={x}^{3}+\left(17a-171\right){x}+256a-186$
448.1-c1 448.1-c \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $5.784310009$ 1.262239926 \( \frac{11664}{49} \) \( \bigl[0\) , \( a + 1\) , \( 0\) , \( -2 a + 11\) , \( 13 a - 33\bigr] \) ${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-2a+11\right){x}+13a-33$
448.1-c2 448.1-c \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $23.13724003$ 1.262239926 \( \frac{55296}{7} \) \( \bigl[0\) , \( a + 1\) , \( 0\) , \( 3 a - 4\) , \( 2 a - 4\bigr] \) ${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(3a-4\right){x}+2a-4$
448.1-c3 448.1-c \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $5.784310009$ 1.262239926 \( -\frac{77586336}{7} a + \frac{216622512}{7} \) \( \bigl[0\) , \( a + 1\) , \( 0\) , \( 33 a - 89\) , \( 143 a - 400\bigr] \) ${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(33a-89\right){x}+143a-400$
448.1-c4 448.1-c \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) 0 $\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $23.13724003$ 1.262239926 \( \frac{77586336}{7} a + \frac{139036176}{7} \) \( \bigl[0\) , \( a + 1\) , \( 0\) , \( 8 a - 24\) , \( -20 a + 56\bigr] \) ${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(8a-24\right){x}-20a+56$
448.1-d1 448.1-d \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) $1$ $\Z/4\Z$ $\mathrm{SU}(2)$ $1.294122201$ $12.23735606$ 3.455836897 \( \frac{432}{7} \) \( \bigl[0\) , \( 0\) , \( 0\) , \( 1\) , \( 2\bigr] \) ${y}^2={x}^{3}+{x}+2$
448.1-d2 448.1-d \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $5.176488807$ $3.059339015$ 3.455836897 \( \frac{11090466}{2401} \) \( \bigl[0\) , \( 0\) , \( 0\) , \( -59\) , \( -138\bigr] \) ${y}^2={x}^{3}-59{x}-138$
448.1-d3 448.1-d \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) $1$ $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $2.588244403$ $12.23735606$ 3.455836897 \( \frac{740772}{49} \) \( \bigl[0\) , \( 0\) , \( 0\) , \( -19\) , \( 30\bigr] \) ${y}^2={x}^{3}-19{x}+30$
448.1-d4 448.1-d \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $1.294122201$ $12.23735606$ 3.455836897 \( \frac{1443468546}{7} \) \( \bigl[0\) , \( 0\) , \( 0\) , \( -299\) , \( 1990\bigr] \) ${y}^2={x}^{3}-299{x}+1990$
448.1-e1 448.1-e \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) $1$ $\Z/4\Z$ $\mathrm{SU}(2)$ $1.891959002$ $5.272587057$ 2.176836612 \( \frac{432}{7} \) \( \bigl[0\) , \( 0\) , \( 0\) , \( 5 a + 9\) , \( -48 a - 86\bigr] \) ${y}^2={x}^{3}+\left(5a+9\right){x}-48a-86$
448.1-e2 448.1-e \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $1.891959002$ $5.272587057$ 2.176836612 \( \frac{11090466}{2401} \) \( \bigl[0\) , \( 0\) , \( 0\) , \( 295 a - 826\) , \( -3312 a + 9246\bigr] \) ${y}^2={x}^{3}+\left(295a-826\right){x}-3312a+9246$
448.1-e3 448.1-e \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) $1$ $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $3.783918005$ $5.272587057$ 2.176836612 \( \frac{740772}{49} \) \( \bigl[0\) , \( 0\) , \( 0\) , \( 95 a - 266\) , \( 720 a - 2010\bigr] \) ${y}^2={x}^{3}+\left(95a-266\right){x}+720a-2010$
448.1-e4 448.1-e \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $7.567836011$ $1.318146764$ 2.176836612 \( \frac{1443468546}{7} \) \( \bigl[0\) , \( 0\) , \( 0\) , \( 1495 a - 4186\) , \( 47760 a - 133330\bigr] \) ${y}^2={x}^{3}+\left(1495a-4186\right){x}+47760a-133330$
448.1-f1 448.1-f \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $5.784310009$ 1.262239926 \( \frac{11664}{49} \) \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 4 a + 8\) , \( -16 a - 28\bigr] \) ${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(4a+8\right){x}-16a-28$
448.1-f2 448.1-f \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $23.13724003$ 1.262239926 \( \frac{55296}{7} \) \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -a - 2\) , \( 0\bigr] \) ${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-a-2\right){x}$
448.1-f3 448.1-f \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) 0 $\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $23.13724003$ 1.262239926 \( -\frac{77586336}{7} a + \frac{216622512}{7} \) \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -6 a - 17\) , \( 27 a + 53\bigr] \) ${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-6a-17\right){x}+27a+53$
448.1-f4 448.1-f \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $5.784310009$ 1.262239926 \( \frac{77586336}{7} a + \frac{139036176}{7} \) \( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -31 a - 57\) , \( -111 a - 200\bigr] \) ${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-31a-57\right){x}-111a-200$
448.1-g1 448.1-g \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) $1$ $\Z/4\Z$ $\mathrm{SU}(2)$ $1.530313332$ $8.919127558$ 2.978469037 \( -\frac{55296}{49} \) \( \bigl[0\) , \( 0\) , \( 0\) , \( -2 a - 4\) , \( -4 a - 7\bigr] \) ${y}^2={x}^{3}+\left(-2a-4\right){x}-4a-7$
448.1-g2 448.1-g \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $1.530313332$ $8.919127558$ 2.978469037 \( -\frac{14919813068184}{7} a + \frac{41645492922276}{7} \) \( \bigl[0\) , \( 0\) , \( 0\) , \( -17 a - 154\) , \( -256 a + 70\bigr] \) ${y}^2={x}^{3}+\left(-17a-154\right){x}-256a+70$
448.1-g3 448.1-g \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) $1$ $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $3.060626665$ $8.919127558$ 2.978469037 \( \frac{21882096}{7} \) \( \bigl[0\) , \( 0\) , \( 0\) , \( -37 a - 74\) , \( -200 a - 350\bigr] \) ${y}^2={x}^{3}+\left(-37a-74\right){x}-200a-350$
448.1-g4 448.1-g \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $6.121253330$ $2.229781889$ 2.978469037 \( \frac{14919813068184}{7} a + \frac{26725679854092}{7} \) \( \bigl[0\) , \( 0\) , \( 0\) , \( -617 a - 1114\) , \( -12688 a - 22722\bigr] \) ${y}^2={x}^{3}+\left(-617a-1114\right){x}-12688a-22722$
448.1-h1 448.1-h \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $3.594960974$ 0.784484799 \( -\frac{4}{7} \) \( \bigl[0\) , \( -1\) , \( 0\) , \( 0\) , \( -4\bigr] \) ${y}^2={x}^{3}-{x}^{2}-4$
448.1-h2 448.1-h \(\Q(\sqrt{21}) \) \( 2^{6} \cdot 7 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $3.594960974$ 0.784484799 \( \frac{3543122}{49} \) \( \bigl[0\) , \( -1\) , \( 0\) , \( -40\) , \( -84\bigr] \) ${y}^2={x}^{3}-{x}^{2}-40{x}-84$
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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.