Elliptic curve with LMFDB label 14896.a2 (Cremona label 14896bg2)

• Label: 14896.a2
• Conductor: $14896$
• Discriminant: $-3.305\times 10^{12}$
• j-invariant: \( -\frac{89915392}{6859} \)
• CM: no
• Rank: $0$
• Torsion structure: trivial

Minimal Weierstrass equation

\(y^2=x^3+x^2-7317x-258749\)  

Mordell-Weil group structure

trivial

Integral points

\(\)  

Invariants

Conductor:	\( 14896 \)	=	\(2^{4} \cdot 7^{2} \cdot 19\)
Discriminant:	\(-3305285595136 \)	=	\(-1 \cdot 2^{12} \cdot 7^{6} \cdot 19^{3} \)
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:	\(1.1502352677448649197923807583\dots\)
Stable Faltings height:	\(-0.51586698734273704217752773488\dots\)

BSD invariants

Analytic rank:	\(0\)
Regulator:	\(1\)
Real period:	\(0.25697043554353832962913884785\dots\)
Tamagawa product:	\( 3 \)  = \( 1\cdot1\cdot3 \)
Torsion order:	\(1\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 14896.2.a.a

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

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

Modular degree:	27216
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L(E,1) \) ≈ \( 0.77091130663061498888741654356109804784 \)

Local data

This elliptic curve is not semistable. There are 3 primes of bad reduction:

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(1\)	\(II^{*}\)	Additive	-1	4	12	0
\(7\)	\(1\)	\(I_0^{*}\)	Additive	-1	2	6	0
\(19\)	\(3\)	\(I_{3}\)	Split multiplicative	-1	1	3	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

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

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	add	ordinary	ordinary	add	ordinary	ordinary	ordinary	split	ss	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary
$\lambda$-invariant(s)	-	2	0	-	0	0	0	1	0,0	0	0	0	0	2	0
$\mu$-invariant(s)	-	1	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 \(d=\) 3.
Its isogeny class 14896.a 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
$2$	\(\Q(\sqrt{7}) \)	\(\Z/3\Z\)	2.2.28.1-361.1-b2
$2$	\(\Q(\sqrt{-21}) \)	\(\Z/3\Z\)	Not in database
$3$	3.1.76.1	\(\Z/2\Z\)	Not in database
$4$	\(\Q(\sqrt{-3}, \sqrt{7})\)	\(\Z/3\Z \times \Z/3\Z\)	Not in database
$6$	6.0.109744.2	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$6$	6.2.31698688.1	\(\Z/6\Z\)	Not in database
$6$	6.0.855864576.2	\(\Z/6\Z\)	Not in database
$12$	Deg 12	\(\Z/4\Z\)	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
$12$	Deg 12	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$18$	18.6.17068408244742026681830076055552.1	\(\Z/9\Z\)	Not in database
$18$	18.0.178541578852998866709992697969426969133056.2	\(\Z/9\Z\)	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.