Elliptic curves over \(\Q(\sqrt{2}, \sqrt{3})\) of conductor 47.2

Results (8 matches)

 

Label	Class	Class size	Class degree	Base field	Field degree	Field signature	Conductor	Conductor norm	Discriminant norm	Root analytic conductor	Bad primes	Rank	Torsion	CM	CM	Sato-Tate	$\Q$-curve	Base change	Semistable	Potentially good	Nonmax $\ell$	mod-$\ell$ images	$Ш_{\textrm{an}}$	Tamagawa	Regulator	Period	Leading coeff	j-invariant	Weierstrass coefficients	Weierstrass equation
47.2-a1	47.2-a	$4$	$4$	\(\Q(\sqrt{2}, \sqrt{3})\)	$4$	$[4, 0]$	47.2	\( 47 \)	\( -47 \)	$6.94052$	$(a^3+2a^2-2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	\( 1 \)	$1$	$420.1222714$	2.188136830	\( \frac{76092676472}{47} a^{3} + \frac{147059917488}{47} a^{2} - \frac{20384319416}{47} a - \frac{39402167512}{47} \)	\( \bigl[a + 1\) , \( -a^{2} + 2\) , \( a^{3} + a^{2} - 4 a - 1\) , \( 3 a^{3} - 2 a^{2} - 15 a - 6\) , \( -7 a^{3} - 5 a^{2} + 15 a + 8\bigr] \)	${y}^2+\left(a+1\right){x}{y}+\left(a^{3}+a^{2}-4a-1\right){y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(3a^{3}-2a^{2}-15a-6\right){x}-7a^{3}-5a^{2}+15a+8$
47.2-a2	47.2-a	$4$	$4$	\(\Q(\sqrt{2}, \sqrt{3})\)	$4$	$[4, 0]$	47.2	\( 47 \)	\( 47^{2} \)	$6.94052$	$(a^3+2a^2-2)$	0	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2Cs	$1$	\( 2 \)	$1$	$840.2445428$	2.188136830	\( \frac{27756544}{2209} a^{3} + \frac{44907520}{2209} a^{2} - \frac{20659712}{2209} a - \frac{948672}{2209} \)	\( \bigl[a^{2} - 1\) , \( a^{3} + a^{2} - 5 a - 1\) , \( a\) , \( -2 a^{3} + 2 a^{2} + 3 a\) , \( -a^{3} + a^{2} + 2 a - 1\bigr] \)	${y}^2+\left(a^{2}-1\right){x}{y}+a{y}={x}^{3}+\left(a^{3}+a^{2}-5a-1\right){x}^{2}+\left(-2a^{3}+2a^{2}+3a\right){x}-a^{3}+a^{2}+2a-1$
47.2-a3	47.2-a	$4$	$4$	\(\Q(\sqrt{2}, \sqrt{3})\)	$4$	$[4, 0]$	47.2	\( 47 \)	\( 47^{4} \)	$6.94052$	$(a^3+2a^2-2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	\( 2 \)	$1$	$210.0611357$	2.188136830	\( -\frac{1194734094856}{4879681} a^{3} - \frac{628684252240}{4879681} a^{2} + \frac{4457544169928}{4879681} a + \frac{2348620171576}{4879681} \)	\( \bigl[a^{3} + a^{2} - 4 a - 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( a^{2} - 2\) , \( 8 a^{3} + 5 a^{2} - 29 a - 15\) , \( 16 a^{3} + 9 a^{2} - 60 a - 32\bigr] \)	${y}^2+\left(a^{3}+a^{2}-4a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}+a^{2}-3a-1\right){x}^{2}+\left(8a^{3}+5a^{2}-29a-15\right){x}+16a^{3}+9a^{2}-60a-32$
47.2-a4	47.2-a	$4$	$4$	\(\Q(\sqrt{2}, \sqrt{3})\)	$4$	$[4, 0]$	47.2	\( 47 \)	\( -47 \)	$6.94052$	$(a^3+2a^2-2)$	0	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	\( 1 \)	$1$	$420.1222714$	2.188136830	\( \frac{706906112}{47} a^{3} - \frac{365769728}{47} a^{2} - \frac{2638573056}{47} a + \frac{1365858240}{47} \)	\( \bigl[a^{2} - 1\) , \( -a^{3} + a^{2} + 5 a - 2\) , \( a\) , \( a^{2} + 3 a\) , \( a^{3}\bigr] \)	${y}^2+\left(a^{2}-1\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+a^{2}+5a-2\right){x}^{2}+\left(a^{2}+3a\right){x}+a^{3}$
47.2-b1	47.2-b	$4$	$4$	\(\Q(\sqrt{2}, \sqrt{3})\)	$4$	$[4, 0]$	47.2	\( 47 \)	\( -47 \)	$6.94052$	$(a^3+2a^2-2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	\( 1 \)	$0.085869761$	$505.3519892$	0.904051146	\( \frac{76092676472}{47} a^{3} + \frac{147059917488}{47} a^{2} - \frac{20384319416}{47} a - \frac{39402167512}{47} \)	\( \bigl[a^{3} + a^{2} - 4 a - 2\) , \( a^{3} + a^{2} - 4 a - 1\) , \( a^{3} - 3 a + 1\) , \( 4 a^{3} - 19 a - 9\) , \( 11 a^{3} + 4 a^{2} - 33 a - 18\bigr] \)	${y}^2+\left(a^{3}+a^{2}-4a-2\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(a^{3}+a^{2}-4a-1\right){x}^{2}+\left(4a^{3}-19a-9\right){x}+11a^{3}+4a^{2}-33a-18$
47.2-b2	47.2-b	$4$	$4$	\(\Q(\sqrt{2}, \sqrt{3})\)	$4$	$[4, 0]$	47.2	\( 47 \)	\( 47^{2} \)	$6.94052$	$(a^3+2a^2-2)$	$1$	$\Z/2\Z\oplus\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2Cs	$1$	\( 2 \)	$0.042934880$	$2021.407957$	0.904051146	\( \frac{27756544}{2209} a^{3} + \frac{44907520}{2209} a^{2} - \frac{20659712}{2209} a - \frac{948672}{2209} \)	\( \bigl[a^{2} - 1\) , \( -a^{3} + a^{2} + 5 a - 2\) , \( a^{3} - 4 a\) , \( 3 a^{3} + a^{2} - 7 a + 1\) , \( a^{3} + a^{2} - a - 1\bigr] \)	${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-2\right){x}^{2}+\left(3a^{3}+a^{2}-7a+1\right){x}+a^{3}+a^{2}-a-1$
47.2-b3	47.2-b	$4$	$4$	\(\Q(\sqrt{2}, \sqrt{3})\)	$4$	$[4, 0]$	47.2	\( 47 \)	\( 47^{4} \)	$6.94052$	$(a^3+2a^2-2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	\( 2^{2} \)	$0.021467440$	$505.3519892$	0.904051146	\( -\frac{1194734094856}{4879681} a^{3} - \frac{628684252240}{4879681} a^{2} + \frac{4457544169928}{4879681} a + \frac{2348620171576}{4879681} \)	\( \bigl[a + 1\) , \( -a^{2} - a + 2\) , \( a\) , \( 7 a^{3} + 3 a^{2} - 27 a - 12\) , \( -9 a^{3} - 5 a^{2} + 34 a + 18\bigr] \)	${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(7a^{3}+3a^{2}-27a-12\right){x}-9a^{3}-5a^{2}+34a+18$
47.2-b4	47.2-b	$4$	$4$	\(\Q(\sqrt{2}, \sqrt{3})\)	$4$	$[4, 0]$	47.2	\( 47 \)	\( -47 \)	$6.94052$	$(a^3+2a^2-2)$	$1$	$\Z/2\Z$	$\textsf{no}$		$\mathrm{SU}(2)$			✓		$2$	2B	$1$	\( 1 \)	$0.021467440$	$2021.407957$	0.904051146	\( \frac{706906112}{47} a^{3} - \frac{365769728}{47} a^{2} - \frac{2638573056}{47} a + \frac{1365858240}{47} \)	\( \bigl[a^{2} - 1\) , \( a^{3} + a^{2} - 5 a - 1\) , \( a^{3} - 4 a\) , \( -3 a^{3} + 2 a^{2} + 9 a - 1\) , \( a^{3} + a^{2} - 5 a - 1\bigr] \)	${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(a^{3}+a^{2}-5a-1\right){x}^{2}+\left(-3a^{3}+2a^{2}+9a-1\right){x}+a^{3}+a^{2}-5a-1$

 

  *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.