Elliptic curve with Cremona label 1710b2 (LMFDB label 1710.h1)

• Label: 1710b2
• Conductor: $1710$
• Discriminant: $2814933600$
• j-invariant: \( \frac{260549802603}{104256800} \)
• CM: no
• Rank: $1$
• Torsion structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\(y^2+xy=x^3-x^2-399x+1805\)

Mordell-Weil group structure

$\Z\times \Z/{2}\Z$

Infinite order Mordell-Weil generator and height

$P$	=	\(\left(19, 19\right)\)
$\hat{h}(P)$	≈	$0.41816026962396187262953016227$

Torsion generators

\( \left(\frac{19}{4}, -\frac{19}{8}\right) \)

Integral points

\( \left(-19, 57\right) \), \( \left(-19, -38\right) \), \( \left(1, 37\right) \), \( \left(1, -38\right) \), \( \left(19, 19\right) \), \( \left(19, -38\right) \), \( \left(95, 855\right) \), \( \left(95, -950\right) \)

Invariants

Conductor:	\( 1710 \)	=	$2 \cdot 3^{2} \cdot 5 \cdot 19$
Discriminant:	$2814933600 $	=	$2^{5} \cdot 3^{3} \cdot 5^{2} \cdot 19^{4} $
j-invariant:	\( \frac{260549802603}{104256800} \)	=	$2^{-5} \cdot 3^{3} \cdot 5^{-2} \cdot 19^{-4} \cdot 2129^{3}$
Endomorphism ring:	$\Z$
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$
Faltings height:	$0.51130370158950393265705933602\dots$
Stable Faltings height:	$0.23665062942247650980824802679\dots$

BSD invariants

Analytic rank:	$1$
Regulator:	$0.41816026962396187262953016227\dots$
Real period:	$1.3011095265311355398600504392\dots$
Tamagawa product:	$ 16 $  = $ 1\cdot2\cdot2\cdot2^{2} $
Torsion order:	$2$
Analytic order of Ш:	$1$ (exact)
Special value:	$ L'(E,1) $ ≈ $ 2.1762892416982600435753104124 $

Modular invariants

Modular form   1710.2.a.h

\( q - q^{2} + q^{4} + q^{5} - 2 q^{7} - q^{8} - q^{10} - 2 q^{11} - 4 q^{13} + 2 q^{14} + q^{16} + 6 q^{17} + q^{19} + O(q^{20}) \)

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

Modular degree:	1280
$ \Gamma_0(N) $-optimal:	no
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$(j)_{-}$
$2$	$1$	$I_{5}$	Non-split multiplicative	1	1	5	5
$3$	$2$	$III$	Additive	1	2	3	0
$5$	$2$	$I_{2}$	Split multiplicative	-1	1	2	2
$19$	$4$	$I_{4}$	Split multiplicative	-1	1	4	4

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

$p$-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	nonsplit	add	split	ord	ord	ord	ord	split	ord	ord	ord	ord	ord	ss	ss
$\lambda$-invariant(s)	2	-	2	1	1	1	1	2	1	1	1	1	1	1,1	1,3
$\mu$-invariant(s)	0	-	0	0	0	0	0	0	0	0	0	0	0	0,0	0,0

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 1710b 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
$2$	\(\Q(\sqrt{6}) \)	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$4$	4.0.21600.1	\(\Z/4\Z\)	Not in database
$8$	8.4.76441190400.11	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	8.0.29859840000.6	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	8.2.2850120270000.5	\(\Z/6\Z\)	Not in database
$16$	Deg 16	\(\Z/8\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/6\Z\)	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.