Elliptic curve with LMFDB label 3249.d2 (Cremona label 3249c2)

• Label: 3249.d2
• Conductor: $3249$
• Discriminant: $-2.352\times 10^{14}$
• j-invariant: \( -\frac{89915392}{6859} \)
• CM: no
• Rank: $1$
• Torsion structure: trivial

Minimal Weierstrass equation

\(y^2+y=x^3-30324x-2162300\)  

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

\(P\)	=	\(\left(\frac{817}{4}, \frac{3245}{8}\right)\)
\(\hat{h}(P)\)	≈	$3.0674714941539549255788330565$

Integral points

\(\)  

Invariants

Conductor:	\( 3249 \)	=	\(3^{2} \cdot 19^{2}\)
Discriminant:	\(-235239331680891 \)	=	\(-1 \cdot 3^{6} \cdot 19^{9} \)
j-invariant:	\( -\frac{89915392}{6859} \)	=	\(-1 \cdot 2^{18} \cdot 7^{3} \cdot 19^{-3}\)
Endomorphism ring:	\(\Z\)
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$
Faltings height:	\(1.5056586465745380335246085995\dots\)
Stable Faltings height:	\(-0.51586698734273704217752773491\dots\)

BSD invariants

Analytic rank:	\(1\)
Regulator:	\(3.0674714941539549255788330565\dots\)
Real period:	\(0.18010457648394080991294602404\dots\)
Tamagawa product:	\( 4 \)  = \( 2\cdot2 \)
Torsion order:	\(1\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form   3249.2.a.d

\( q - 2q^{4} - 3q^{5} - q^{7} - 3q^{11} + 4q^{13} + 4q^{16} + 3q^{17} + O(q^{20}) \)  

For more coefficients, see the Downloads section to the right.

Modular degree:	8640
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L'(E,1) \) ≈ \( 2.2098626173246366794409076156276844651 \)

Local data

This elliptic curve is not semistable. There are 2 primes of bad reduction:

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(3\)	\(2\)	\(I_0^{*}\)	Additive	-1	2	6	0
\(19\)	\(2\)	\(I_3^{*}\)	Additive	-1	2	9	3

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime	Image of Galois representation
\(3\)	Cs

$p$-adic data

$p$-adic regulators

\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	ss	add	ordinary	ordinary	ordinary	ordinary	ordinary	add	ss	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary
$\lambda$-invariant(s)	?	-	1	1	1	1	1	-	1,1	1	1	1	1	1	1
$\mu$-invariant(s)	?	-	0	0	0	0	0	-	0,0	0	0	0	0	0	0

An entry ? indicates that the invariants have not yet been computed.

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 3249.d consists of 2 curves linked by isogenies of degrees dividing 9.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$2$	\(\Q(\sqrt{57}) \)	\(\Z/3\Z\)	2.2.57.1-19.1-a2
$2$	\(\Q(\sqrt{-19}) \)	\(\Z/3\Z\)	Not in database
$3$	3.1.76.1	\(\Z/2\Z\)	Not in database
$4$	\(\Q(\sqrt{-3}, \sqrt{-19})\)	\(\Z/3\Z \times \Z/3\Z\)	Not in database
$6$	6.0.109744.2	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$6$	6.2.2963088.1	\(\Z/6\Z\)	Not in database
$12$	Deg 12	\(\Z/4\Z\)	Not in database
$12$	12.0.8779890495744.1	\(\Z/6\Z \times \Z/6\Z\)	Not in database
$18$	18.6.298810123329025654919217.1	\(\Z/9\Z\)	Not in database
$18$	18.0.38588388032760475707449435212371.2	\(\Z/9\Z\)	Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.