Elliptic curve with LMFDB label 15680.v1 (Cremona label 15680df2)

• Label: 15680.v1
• Conductor: $15680$
• Discriminant: $-157351936000$
• j-invariant: \( -\frac{5452947409}{250} \)
• CM: no
• Rank: $1$
• Torsion structure: trivial

Minimal Weierstrass equation

\(y^2=x^3+x^2-31425x+2133823\)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

\(P\)	=	\( \left(111, 160\right) \)
\(\hat{h}(P)\)	≈	$0.30982881349445581663467229175$

Integral points

\((-17,\pm 1632)\), \((101,\pm 20)\), \((111,\pm 160)\)

Invariants

Conductor:	\( 15680 \)	=	\(2^{6} \cdot 5 \cdot 7^{2}\)
Discriminant:	\(-157351936000 \)	=	\(-1 \cdot 2^{19} \cdot 5^{3} \cdot 7^{4} \)
j-invariant:	\( -\frac{5452947409}{250} \)	=	\(-1 \cdot 2^{-1} \cdot 5^{-3} \cdot 7^{2} \cdot 13^{3} \cdot 37^{3}\)
Endomorphism ring:	\(\Z\)
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$

BSD invariants

Analytic rank:	\(1\)
Regulator:	\(0.30982881349445581663467229175\)
Real period:	\(0.96434782079473871775803435719\)
Tamagawa product:	\( 12 \)  = \( 2^{2}\cdot3\cdot1 \)
Torsion order:	\(1\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 15680.2.a.v

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

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

Modular degree:	34560
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L'(E,1) \) ≈ \( 3.5853928933535760338285153851943643683 \)

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\((j)_{-}\)
\(2\)	\(4\)	\(I_9^{*}\)	Additive	-1	6	19	1
\(5\)	\(3\)	\(I_{3}\)	Split multiplicative	-1	1	3	3
\(7\)	\(1\)	\(IV\)	Additive	1	2	4	0

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
\(3\)	B

$p$-adic data

$p$-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	add	ordinary	split	add	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary
$\lambda$-invariant(s)	-	3	2	-	1	1	1	1	1	1	1	1	1	1	1
$\mu$-invariant(s)	-	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 15680.v consists of 2 curves linked by isogenies of degree 3.

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
$2$	\(\Q(\sqrt{6}) \)	\(\Z/3\Z\)	Not in database
$3$	3.1.1960.1	\(\Z/2\Z\)	Not in database
$6$	6.0.153664000.2	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$6$	6.0.3584673792.8	\(\Z/3\Z\)	Not in database
$6$	6.2.829785600.1	\(\Z/6\Z\)	Not in database
$12$	Deg 12	\(\Z/4\Z\)	Not in database
$12$	Deg 12	\(\Z/3\Z \times \Z/3\Z\)	Not in database
$12$	Deg 12	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$18$	18.6.55337594319760708207319187456000000000000.1	\(\Z/9\Z\)	Not in database
$18$	18.0.719728910524754422652731392000000.1	\(\Z/6\Z\)	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.