Elliptic Curve with Cremona label 19890r1 (LMFDB label 19890.n7)

• Label: 19890r1
• Conductor: 19890
• Discriminant: 31277723392666710835200
• j-invariant: \( \frac{1018563973439611524445729}{42904970360310988800} \)
• CM: no
• Rank: 0
• Torsion Structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\( y^2 + x y = x^{3} - x^{2} - 18865314 x - 30364499052 \)

Mordell-Weil group structure

\(\Z/{2}\Z\)

Torsion generators

\( \left(-2892, 1446\right) \)

Integral points

\( \left(-2892, 1446\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

Conductor:	\( 19890 \)	=	\(2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17\)
Discriminant:	\(31277723392666710835200 \)	=	\(2^{24} \cdot 3^{12} \cdot 5^{2} \cdot 13^{4} \cdot 17^{3} \)
j-invariant:	\( \frac{1018563973439611524445729}{42904970360310988800} \)	=	\(2^{-24} \cdot 3^{-6} \cdot 5^{-2} \cdot 11^{6} \cdot 13^{-4} \cdot 17^{-3} \cdot 831529^{3}\)
Endomorphism ring:	\(\Z\)		(no Complex Multiplication)
Sato-Tate Group:	$\mathrm{SU}(2)$

BSD invariants

Rank:	\(0\)
Regulator:	\(1\)
Real period:	\(0.0726292685095\)
Tamagawa product:	\( 32 \)  = \( 2\cdot2\cdot2\cdot2^{2}\cdot1 \)
Torsion order:	\(2\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 19890.2.a.n

\( q - q^{2} + q^{4} + q^{5} - 4q^{7} - q^{8} - q^{10} + q^{13} + 4q^{14} + q^{16} - q^{17} - 4q^{19} + O(q^{20}) \)

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

Modular degree:	2211840
\( \Gamma_0(N) \)-optimal:	yes
Manin constant:	1

Special L-value

\( L(E,1) \) ≈ \( 0.581034148076 \)

Local data

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(2\)	\( I_{24} \)	Non-split multiplicative	1	1	24	24
\(3\)	\(2\)	\( I_6^{*} \)	Additive	-1	2	12	6
\(5\)	\(2\)	\( I_{2} \)	Split multiplicative	-1	1	2	2
\(13\)	\(4\)	\( I_{4} \)	Split multiplicative	-1	1	4	4
\(17\)	\(1\)	\( I_{3} \)	Non-split multiplicative	1	1	3	3

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X13.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ and has index 6.

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

prime	Image of Galois representation
\(2\)	B
\(3\)	B.1.2

$p$-adic data

$p$-adic regulators

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$	2	3	5	13	17
Reduction type	nonsplit	add	split	split	nonsplit
$\lambda$-invariant(s)	6	-	1	1	0
$\mu$-invariant(s)	0	-	0	0	0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

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 \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 19890r consists of 8 curves linked by isogenies of degrees dividing 12.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 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	\(\Q(\sqrt{17}) \)	\(\Z/2\Z \times \Z/2\Z\)	Not in database
	\(\Q(\sqrt{-51}) \)	\(\Z/4\Z\)	Not in database
	\(\Q(\sqrt{-3}) \)	\(\Z/12\Z\)	Not in database
3	3.1.12675.1	\(\Z/6\Z\)	Not in database
4	\(\Q(\sqrt{-3}, \sqrt{17})\)	\(\Z/2\Z \times \Z/12\Z\)	Not in database
6	6.0.481966875.1	\(\Z/3\Z \times \Z/12\Z\)	Not in database
	6.0.2367903256875.1	\(\Z/12\Z\)	Not in database
	6.2.789301085625.1	\(\Z/2\Z \times \Z/6\Z\)	Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.