Elliptic curves over \(\Q(\sqrt{105}) \) of conductor 35.1

The results below are complete, since the LMFDB contains all elliptic curves with conductor norm at most 5000 over real quadratic fields with discriminant 497

Results (12 matches)

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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
35.1-a1	35.1-a	$3$	$9$	\(\Q(\sqrt{105}) \)	$2$	$[2, 0]$	35.1	\( 5 \cdot 7 \)	\( 5^{18} \cdot 7^{2} \)	$2.22715$	$(2a-11), (7,a+3)$	$2$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3B	$1$	\( 2^{2} \)	$3.380268306$	$0.494084210$	2.607819218	\( -\frac{250523582464}{13671875} \)	\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -86155 a - 398325\) , \( -32460825 a - 150081819\bigr] \)	${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-86155a-398325\right){x}-32460825a-150081819$
35.1-a2	35.1-a	$3$	$9$	\(\Q(\sqrt{105}) \)	$2$	$[2, 0]$	35.1	\( 5 \cdot 7 \)	\( 5^{2} \cdot 7^{2} \)	$2.22715$	$(2a-11), (7,a+3)$	$2$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3B	$1$	\( 2^{2} \)	$0.041731707$	$40.02082101$	2.607819218	\( -\frac{262144}{35} \)	\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -875 a - 4035\) , \( 36705 a + 169711\bigr] \)	${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-875a-4035\right){x}+36705a+169711$
35.1-a3	35.1-a	$3$	$9$	\(\Q(\sqrt{105}) \)	$2$	$[2, 0]$	35.1	\( 5 \cdot 7 \)	\( 5^{6} \cdot 7^{6} \)	$2.22715$	$(2a-11), (7,a+3)$	$2$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3Cs	$1$	\( 2^{2} \)	$0.375585367$	$4.446757890$	2.607819218	\( \frac{71991296}{42875} \)	\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( 5685 a + 26295\) , \( -98901 a - 457260\bigr] \)	${y}^2+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(5685a+26295\right){x}-98901a-457260$
35.1-b1	35.1-b	$3$	$9$	\(\Q(\sqrt{105}) \)	$2$	$[2, 0]$	35.1	\( 5 \cdot 7 \)	\( 2^{12} \cdot 5^{18} \cdot 7^{2} \)	$2.22715$	$(2a-11), (7,a+3)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3B.1.1	$1$	\( 2^{2} \cdot 3^{2} \)	$1$	$4.862220259$	1.898016442	\( -\frac{250523582464}{13671875} \)	\( \bigl[0\) , \( 1\) , \( a\) , \( -12214 a - 56473\) , \( 1735781 a + 8025336\bigr] \)	${y}^2+a{y}={x}^{3}+{x}^{2}+\left(-12214a-56473\right){x}+1735781a+8025336$
35.1-b2	35.1-b	$3$	$9$	\(\Q(\sqrt{105}) \)	$2$	$[2, 0]$	35.1	\( 5 \cdot 7 \)	\( 2^{12} \cdot 5^{2} \cdot 7^{2} \)	$2.22715$	$(2a-11), (7,a+3)$	0	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3B.1.2	$1$	\( 2^{2} \)	$1$	$4.862220259$	1.898016442	\( -\frac{262144}{35} \)	\( \bigl[0\) , \( 1\) , \( a\) , \( -124 a - 573\) , \( -1929 a - 8924\bigr] \)	${y}^2+a{y}={x}^{3}+{x}^{2}+\left(-124a-573\right){x}-1929a-8924$
35.1-b3	35.1-b	$3$	$9$	\(\Q(\sqrt{105}) \)	$2$	$[2, 0]$	35.1	\( 5 \cdot 7 \)	\( 2^{12} \cdot 5^{6} \cdot 7^{6} \)	$2.22715$	$(2a-11), (7,a+3)$	0	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3Cs.1.1	$1$	\( 2^{2} \cdot 3^{2} \)	$1$	$4.862220259$	1.898016442	\( \frac{71991296}{42875} \)	\( \bigl[0\) , \( 1\) , \( a\) , \( 806 a + 3727\) , \( 5080 a + 23482\bigr] \)	${y}^2+a{y}={x}^{3}+{x}^{2}+\left(806a+3727\right){x}+5080a+23482$
35.1-c1	35.1-c	$3$	$9$	\(\Q(\sqrt{105}) \)	$2$	$[2, 0]$	35.1	\( 5 \cdot 7 \)	\( 2^{12} \cdot 5^{18} \cdot 7^{2} \)	$2.22715$	$(2a-11), (7,a+3)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3B	$1$	\( 2^{2} \cdot 3^{2} \)	$0.040819755$	$4.862220259$	1.394578218	\( -\frac{250523582464}{13671875} \)	\( \bigl[0\) , \( a\) , \( a\) , \( 1445 a - 8134\) , \( -73135 a + 411269\bigr] \)	${y}^2+a{y}={x}^{3}+a{x}^{2}+\left(1445a-8134\right){x}-73135a+411269$
35.1-c2	35.1-c	$3$	$9$	\(\Q(\sqrt{105}) \)	$2$	$[2, 0]$	35.1	\( 5 \cdot 7 \)	\( 2^{12} \cdot 5^{2} \cdot 7^{2} \)	$2.22715$	$(2a-11), (7,a+3)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3B	$1$	\( 2^{2} \)	$0.367377802$	$4.862220259$	1.394578218	\( -\frac{262144}{35} \)	\( \bigl[0\) , \( a\) , \( a\) , \( 15 a - 74\) , \( 55 a - 311\bigr] \)	${y}^2+a{y}={x}^{3}+a{x}^{2}+\left(15a-74\right){x}+55a-311$
35.1-c3	35.1-c	$3$	$9$	\(\Q(\sqrt{105}) \)	$2$	$[2, 0]$	35.1	\( 5 \cdot 7 \)	\( 2^{12} \cdot 5^{6} \cdot 7^{6} \)	$2.22715$	$(2a-11), (7,a+3)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3Cs	$1$	\( 2^{2} \cdot 3 \)	$0.122459267$	$4.862220259$	1.394578218	\( \frac{71991296}{42875} \)	\( \bigl[0\) , \( a\) , \( a\) , \( -95 a + 546\) , \( -48 a + 271\bigr] \)	${y}^2+a{y}={x}^{3}+a{x}^{2}+\left(-95a+546\right){x}-48a+271$
35.1-d1	35.1-d	$3$	$9$	\(\Q(\sqrt{105}) \)	$2$	$[2, 0]$	35.1	\( 5 \cdot 7 \)	\( 5^{18} \cdot 7^{2} \)	$2.22715$	$(2a-11), (7,a+3)$	$1$	$\mathsf{trivial}$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3B.1.2	$1$	\( 2^{2} \)	$7.121947793$	$0.494084210$	2.747230492	\( -\frac{250523582464}{13671875} \)	\( \bigl[0\) , \( 1\) , \( 1\) , \( -131\) , \( -650\bigr] \)	${y}^2+{y}={x}^{3}+{x}^{2}-131{x}-650$
35.1-d2	35.1-d	$3$	$9$	\(\Q(\sqrt{105}) \)	$2$	$[2, 0]$	35.1	\( 5 \cdot 7 \)	\( 5^{2} \cdot 7^{2} \)	$2.22715$	$(2a-11), (7,a+3)$	$1$	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3B.1.1	$1$	\( 2^{2} \)	$0.791327532$	$40.02082101$	2.747230492	\( -\frac{262144}{35} \)	\( \bigl[0\) , \( 1\) , \( 1\) , \( -1\) , \( 0\bigr] \)	${y}^2+{y}={x}^{3}+{x}^{2}-{x}$
35.1-d3	35.1-d	$3$	$9$	\(\Q(\sqrt{105}) \)	$2$	$[2, 0]$	35.1	\( 5 \cdot 7 \)	\( 5^{6} \cdot 7^{6} \)	$2.22715$	$(2a-11), (7,a+3)$	$1$	$\Z/3\Z$	$\textsf{no}$		$\mathrm{SU}(2)$	✓	✓	✓		$3$	3Cs.1.1	$1$	\( 2^{2} \cdot 3 \)	$2.373982597$	$4.446757890$	2.747230492	\( \frac{71991296}{42875} \)	\( \bigl[0\) , \( 1\) , \( 1\) , \( 9\) , \( 1\bigr] \)	${y}^2+{y}={x}^{3}+{x}^{2}+9{x}+1$

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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.