Elliptic curve with Cremona label 32487r1 (LMFDB label 32487.j2)

• Label: 32487r1
• Conductor: $32487$
• Discriminant: $246406065633$
• j-invariant: \( \frac{2000852317801}{2094417} \)
• CM: no
• Rank: $0$
• Torsion structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\(y^2+xy+y=x^3-12864x+559969\)

(, )

Mordell-Weil group structure

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

Torsion generators

\( \left(67, -34\right) \)

Integral points

\( \left(67, -34\right) \)

Invariants

Conductor:	\( 32487 \)	=	$3 \cdot 7^{2} \cdot 13 \cdot 17$
Discriminant:	$246406065633 $	=	$3^{6} \cdot 7^{6} \cdot 13^{2} \cdot 17 $
j-invariant:	\( \frac{2000852317801}{2094417} \)	=	$3^{-6} \cdot 13^{-2} \cdot 17^{-1} \cdot 12601^{3}$
Endomorphism ring:	$\Z$
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$
Faltings height:	$1.1031256507726198704546339859\dots$
Stable Faltings height:	$0.13017057624496321790195761418\dots$

BSD invariants

Analytic rank:	$0$
Regulator:	$1$
Real period:	$0.98244973529654358317991518370\dots$
Tamagawa product:	$ 24 $  = $ ( 2 \cdot 3 )\cdot2\cdot2\cdot1 $
Torsion order:	$2$
Analytic order of Ш:	$1$ (exact)
Special value:	$ L(E,1) $ ≈ $ 5.8946984117792614990794911022 $

Modular invariants

Modular form 32487.2.a.j

\( q + q^{2} + q^{3} - q^{4} + 4 q^{5} + q^{6} - 3 q^{8} + q^{9} + 4 q^{10} + 6 q^{11} - q^{12} + q^{13} + 4 q^{15} - q^{16} - q^{17} + q^{18} - 4 q^{19} + O(q^{20}) \)

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

Modular degree:	103680
$ \Gamma_0(N) $-optimal:	yes
Manin constant:	1

Local data

This elliptic curve is not semistable. There are 4 primes of bad reduction:

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord($N$)	ord($\Delta$)	ord$(j)_{-}$
$3$	$6$	$I_{6}$	Split multiplicative	-1	1	6	6
$7$	$2$	$I_0^{*}$	Additive	-1	2	6	0
$13$	$2$	$I_{2}$	Split multiplicative	-1	1	2	2
$17$	$1$	$I_{1}$	Non-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

The image of the adelic Galois representation has level $884$, index $12$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 881 & 4 \\ 880 & 5 \end{array}\right),\left(\begin{array}{rr} 210 & 1 \\ 779 & 0 \end{array}\right),\left(\begin{array}{rr} 613 & 4 \\ 342 & 9 \end{array}\right),\left(\begin{array}{rr} 665 & 222 \\ 220 & 663 \end{array}\right)$

$p$-adic regulators

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

Iwasawa invariants

$p$	2	3	7	13	17
Reduction type	ord	split	add	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.
Its isogeny class 32487r consists of 2 curves linked by isogenies of degree 2.

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
$2$	\(\Q(\sqrt{17}) \)	\(\Z/2\Z \oplus \Z/2\Z\)	Not in database
$4$	4.4.140777.1	\(\Z/4\Z\)	Not in database
$8$	deg 8	\(\Z/2\Z \oplus \Z/4\Z\)	Not in database
$8$	8.8.5727449317681.2	\(\Z/2\Z \oplus \Z/4\Z\)	Not in database
$8$	deg 8	\(\Z/6\Z\)	Not in database
$16$	deg 16	\(\Z/8\Z\)	Not in database
$16$	deg 16	\(\Z/2\Z \oplus \Z/6\Z\)	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.