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Results (16 matches)

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Label Class Base field Conductor norm Rank Torsion CM Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
4.1-a1 4.1-a \(\Q(\sqrt{57}) \) \( 2^{2} \) $0$ $\mathsf{trivial}$ $1$ $7.662151750$ 1.014876791 \( -\frac{293180476215589246298781}{8} a - \frac{960141789432972248487021}{8} \) \( \bigl[1\) , \( -a + 1\) , \( a\) , \( 71395 a - 305260\) , \( -20071186 a + 85802856\bigr] \) ${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(71395a-305260\right){x}-20071186a+85802856$
4.1-a2 4.1-a \(\Q(\sqrt{57}) \) \( 2^{2} \) $0$ $\mathsf{trivial}$ $1$ $7.662151750$ 1.014876791 \( \frac{293180476215589246298781}{8} a - \frac{626661132824280747392901}{4} \) \( \bigl[1\) , \( a\) , \( a + 1\) , \( -71396 a - 233865\) , \( 20071185 a + 65731670\bigr] \) ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-71396a-233865\right){x}+20071185a+65731670$
4.1-a3 4.1-a \(\Q(\sqrt{57}) \) \( 2^{2} \) $0$ $\mathsf{trivial}$ $1$ $7.662151750$ 1.014876791 \( -\frac{699691689}{2097152} a - \frac{307208349}{2097152} \) \( \bigl[1\) , \( -a + 1\) , \( a\) , \( -215 a + 920\) , \( 2686 a - 11488\bigr] \) ${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-215a+920\right){x}+2686a-11488$
4.1-a4 4.1-a \(\Q(\sqrt{57}) \) \( 2^{2} \) $0$ $\mathsf{trivial}$ $1$ $7.662151750$ 1.014876791 \( \frac{699691689}{2097152} a - \frac{503450019}{1048576} \) \( \bigl[1\) , \( a\) , \( a + 1\) , \( 214 a + 705\) , \( -2687 a - 8802\bigr] \) ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(214a+705\right){x}-2687a-8802$
4.1-b1 4.1-b \(\Q(\sqrt{57}) \) \( 2^{2} \) $0$ $\Z/6\Z$ $1$ $37.29397324$ 2.469853714 \( -\frac{20297286875}{64} a + \frac{43383068421}{32} \) \( \bigl[1\) , \( -a\) , \( a\) , \( 91 a - 386\) , \( -817 a + 3486\bigr] \) ${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(91a-386\right){x}-817a+3486$
4.1-b2 4.1-b \(\Q(\sqrt{57}) \) \( 2^{2} \) $0$ $\Z/2\Z$ $1$ $37.29397324$ 2.469853714 \( -\frac{489}{4} a + 1841 \) \( \bigl[1\) , \( -a\) , \( a\) , \( a - 1\) , \( -7\bigr] \) ${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(a-1\right){x}-7$
4.1-b3 4.1-b \(\Q(\sqrt{57}) \) \( 2^{2} \) $0$ $\Z/2\Z$ $1$ $37.29397324$ 2.469853714 \( \frac{489}{4} a + \frac{6875}{4} \) \( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -2 a\) , \( -a - 7\bigr] \) ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}-2a{x}-a-7$
4.1-b4 4.1-b \(\Q(\sqrt{57}) \) \( 2^{2} \) $0$ $\Z/6\Z$ $1$ $37.29397324$ 2.469853714 \( \frac{20297286875}{64} a + \frac{66468849967}{64} \) \( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -92 a - 295\) , \( 816 a + 2669\bigr] \) ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-92a-295\right){x}+816a+2669$
4.1-c1 4.1-c \(\Q(\sqrt{57}) \) \( 2^{2} \) $0$ $\Z/2\Z$ $1$ $3.156724381$ 0.209059179 \( -\frac{20297286875}{64} a + \frac{43383068421}{32} \) \( \bigl[1\) , \( -a - 1\) , \( a\) , \( -2960 a - 9690\) , \( -168414 a - 551544\bigr] \) ${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2960a-9690\right){x}-168414a-551544$
4.1-c2 4.1-c \(\Q(\sqrt{57}) \) \( 2^{2} \) $0$ $\Z/6\Z$ $1$ $28.41051943$ 0.209059179 \( -\frac{489}{4} a + 1841 \) \( \bigl[1\) , \( -a - 1\) , \( a\) , \( 150 a + 495\) , \( -1331 a - 4361\bigr] \) ${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(150a+495\right){x}-1331a-4361$
4.1-c3 4.1-c \(\Q(\sqrt{57}) \) \( 2^{2} \) $0$ $\Z/6\Z$ $1$ $28.41051943$ 0.209059179 \( \frac{489}{4} a + \frac{6875}{4} \) \( \bigl[1\) , \( a + 1\) , \( a\) , \( -149 a + 645\) , \( 1181 a - 5047\bigr] \) ${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-149a+645\right){x}+1181a-5047$
4.1-c4 4.1-c \(\Q(\sqrt{57}) \) \( 2^{2} \) $0$ $\Z/2\Z$ $1$ $3.156724381$ 0.209059179 \( \frac{20297286875}{64} a + \frac{66468849967}{64} \) \( \bigl[1\) , \( a + 1\) , \( a\) , \( 2961 a - 12650\) , \( 171374 a - 732608\bigr] \) ${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2961a-12650\right){x}+171374a-732608$
4.1-d1 4.1-d \(\Q(\sqrt{57}) \) \( 2^{2} \) $0$ $\mathsf{trivial}$ $1$ $0.077866687$ 1.516113114 \( -\frac{293180476215589246298781}{8} a - \frac{960141789432972248487021}{8} \) \( \bigl[1\) , \( a\) , \( a\) , \( -642629 a - 2104558\) , \( -545795629 a - 1787435506\bigr] \) ${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-642629a-2104558\right){x}-545795629a-1787435506$
4.1-d2 4.1-d \(\Q(\sqrt{57}) \) \( 2^{2} \) $0$ $\mathsf{trivial}$ $1$ $0.077866687$ 1.516113114 \( \frac{293180476215589246298781}{8} a - \frac{626661132824280747392901}{4} \) \( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 642628 a - 2747187\) , \( 545795628 a - 2333231135\bigr] \) ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(642628a-2747187\right){x}+545795628a-2333231135$
4.1-d3 4.1-d \(\Q(\sqrt{57}) \) \( 2^{2} \) $0$ $\Z/7\Z$ $1$ $3.815467665$ 1.516113114 \( -\frac{699691689}{2097152} a - \frac{307208349}{2097152} \) \( \bigl[1\) , \( a\) , \( a\) , \( -239 a - 778\) , \( -6021 a - 19722\bigr] \) ${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-239a-778\right){x}-6021a-19722$
4.1-d4 4.1-d \(\Q(\sqrt{57}) \) \( 2^{2} \) $0$ $\Z/7\Z$ $1$ $3.815467665$ 1.516113114 \( \frac{699691689}{2097152} a - \frac{503450019}{1048576} \) \( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 238 a - 1017\) , \( 6020 a - 25743\bigr] \) ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(238a-1017\right){x}+6020a-25743$
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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.