Elliptic curve with Cremona label 129360gd2 (LMFDB label 129360.eq1)

• Label: 129360gd2
• Conductor: $129360$
• Discriminant: $-2.114\times 10^{26}$
• j-invariant: \( -\frac{2126464142970105856}{438611057788643355} \)
• CM: no
• Rank: $0$
• Torsion structure: trivial

Minimal Weierstrass equation

\(y^2=x^3+x^2-21003621x-700462254381\)

Mordell-Weil group structure

trivial

Integral points

None

Invariants

Conductor:	\( 129360 \)	=	\(2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11\)
Discriminant:	\(-211362415975530914088529920 \)	=	\(-1 \cdot 2^{12} \cdot 3 \cdot 5 \cdot 7^{7} \cdot 11^{15} \)
j-invariant:	\( -\frac{2126464142970105856}{438611057788643355} \)	=	\(-1 \cdot 2^{12} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-1} \cdot 11^{-15} \cdot 179^{3} \cdot 449^{3}\)
Endomorphism ring:	\(\Z\)
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$

BSD invariants

Analytic rank:	\(0\)
Regulator:	\(1\)
Real period:	\(0.025072448596868652289407224786\)
Tamagawa product:	\( 4 \)  = \( 1\cdot1\cdot1\cdot2^{2}\cdot1 \)
Torsion order:	\(1\)
Analytic order of Ш:	\(25\) = $5^2$ (exact)

Modular invariants

Modular form 129360.2.a.eq

\( q + q^{3} - q^{5} + q^{9} - q^{11} + 6q^{13} - q^{15} + 7q^{17} - 5q^{19} + O(q^{20}) \)

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

Modular degree:	57600000
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L(E,1) \) ≈ \( 2.5072448596868652289407224786394761662 \)

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\((j)_{-}\)
\(2\)	\(1\)	\(II^{*}\)	Additive	-1	4	12	0
\(3\)	\(1\)	\(I_{1}\)	Split multiplicative	-1	1	1	1
\(5\)	\(1\)	\(I_{1}\)	Non-split multiplicative	1	1	1	1
\(7\)	\(4\)	\(I_1^{*}\)	Additive	-1	2	7	1
\(11\)	\(1\)	\(I_{15}\)	Non-split multiplicative	1	1	15	15

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
\(5\)	B.4.2

$p$-adic data

$p$-adic regulators

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

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	add	split	nonsplit	add	nonsplit	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary
$\lambda$-invariant(s)	-	3	2	-	0	0	0	0	2	0	0	0	0	0	0
$\mu$-invariant(s)	-	0	1	-	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 \(d=\) 5.
Its isogeny class 129360gd consists of 2 curves linked by isogenies of degree 5.

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
$3$	3.1.4620.1	\(\Z/2\Z\)	Not in database
$4$	4.0.98000.1	\(\Z/5\Z\)	Not in database
$6$	6.0.24652782000.1	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$8$	Deg 8	\(\Z/3\Z\)	Not in database
$10$	10.2.1034218810652343750000000000.1	\(\Z/5\Z\)	Not in database
$12$	Deg 12	\(\Z/4\Z\)	Not in database
$12$	Deg 12	\(\Z/10\Z\)	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.