Elliptic curve with LMFDB label 189618.br2 (Cremona label 189618j4)

• Label: 189618.br2
• Conductor: $189618$
• Discriminant: $-8.527\times 10^{27}$
• j-invariant: \( -\frac{2830680648734534916567625}{1766676274677722124288} \)
• CM: no
• Rank: $0$
• Torsion structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\(y^2+xy=x^3-498053228x-6167895603312\)

Mordell-Weil group structure

$\Z/{2}\Z$

Torsion generators

\( \left(\frac{107839}{4}, -\frac{107839}{8}\right) \)

Integral points

None

Invariants

Conductor:	\( 189618 \)	=	$2 \cdot 3 \cdot 11 \cdot 13^{2} \cdot 17$
Discriminant:	$-8527408942700901249012436992 $	=	$-1 \cdot 2^{12} \cdot 3^{2} \cdot 11^{2} \cdot 13^{18} \cdot 17 $
j-invariant:	\( -\frac{2830680648734534916567625}{1766676274677722124288} \)	=	$-1 \cdot 2^{-12} \cdot 3^{-2} \cdot 5^{3} \cdot 7^{3} \cdot 11^{-2} \cdot 13^{-12} \cdot 17^{-1} \cdot 4041683^{3}$
Endomorphism ring:	$\Z$
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$
Faltings height:	$4.0611140885299172870306205522\dots$
Stable Faltings height:	$2.7786394097991489190038768314\dots$

BSD invariants

Analytic rank:	$0$
Regulator:	$1$
Real period:	$0.015539314208347085820905596224\dots$
Tamagawa product:	$ 192 $  = $ ( 2^{2} \cdot 3 )\cdot2\cdot2\cdot2^{2}\cdot1 $
Torsion order:	$2$
Analytic order of Ш:	$9$ = $3^2$ (exact)
Special value:	$ L(E,1) $ ≈ $ 6.7129837380059410746312175688 $

Modular invariants

Modular form 189618.2.a.br

\( q + q^{2} + q^{3} + q^{4} + q^{6} + 4 q^{7} + q^{8} + q^{9} - q^{11} + q^{12} + 4 q^{14} + q^{16} + q^{17} + q^{18} - 2 q^{19} + O(q^{20}) \)

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

Modular degree:	153280512
$ \Gamma_0(N) $-optimal:	no
Manin constant:	1

Local data

This elliptic curve is not semistable. There are 5 primes of bad reduction:

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord($N$)	ord($\Delta$)	ord$(j)_{-}$
$2$	$12$	$I_{12}$	Split multiplicative	-1	1	12	12
$3$	$2$	$I_{2}$	Split multiplicative	-1	1	2	2
$11$	$2$	$I_{2}$	Non-split multiplicative	1	1	2	2
$13$	$4$	$I_{12}^{*}$	Additive	1	2	18	12
$17$	$1$	$I_{1}$	Split multiplicative	-1	1	1	1

Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$	mod-$\ell$ image	$\ell$-adic image
$2$	2B	2.3.0.1
$3$	3B	3.4.0.1

$p$-adic regulators

All $p$-adic regulators are identically $1$ since the rank is $0$.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 189618.br consists of 4 curves linked by isogenies of degrees dividing 6.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$2$	\(\Q(\sqrt{-39}) \)	\(\Z/6\Z\)	Not in database
$2$	\(\Q(\sqrt{-17}) \)	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$4$	4.2.12514788.1	\(\Z/4\Z\)	Not in database
$4$	\(\Q(\sqrt{-17}, \sqrt{-39})\)	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$6$	6.2.158638659079147533.1	\(\Z/6\Z\)	Not in database
$8$	Deg 8	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	Deg 8	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	Deg 8	\(\Z/12\Z\)	Not in database
$12$	Deg 12	\(\Z/3\Z \times \Z/6\Z\)	Not in database
$12$	Deg 12	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$16$	Deg 16	\(\Z/8\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/12\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/12\Z\)	Not in database
$18$	18.0.193643079756895602999529258711396024499161761922904064.1	\(\Z/18\Z\)	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.