Elliptic curve with Cremona label 3025g1 (LMFDB label 3025.a3)

• Label: 3025g1
• Conductor: $3025$
• Discriminant: $-304487046875$
• j-invariant: \( -\frac{4096}{11} \)
• CM: no
• Rank: $0$
• Torsion structure: trivial

Minimal Weierstrass equation

\(y^2+y=x^3+x^2-1008x-29606\)  

Mordell-Weil group structure

trivial

Integral points

\(\)  

Invariants

Conductor:	\( 3025 \)	=	\(5^{2} \cdot 11^{2}\)
Discriminant:	\(-304487046875 \)	=	\(-1 \cdot 5^{6} \cdot 11^{7} \)
j-invariant:	\( -\frac{4096}{11} \)	=	\(-1 \cdot 2^{12} \cdot 11^{-1}\)
Endomorphism ring:	\(\Z\)
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$
Faltings height:	\(0.89093779528078220734195752752\dots\)
Stable Faltings height:	\(-1.1127287973354532519893939281\dots\)

BSD invariants

Analytic rank:	\(0\)
Regulator:	\(1\)
Real period:	\(0.39341358499961023967367211512\dots\)
Tamagawa product:	\( 2 \)  = \( 1\cdot2 \)
Torsion order:	\(1\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form   3025.2.a.a

\( q - 2q^{2} + q^{3} + 2q^{4} - 2q^{6} - 2q^{7} - 2q^{9} + 2q^{12} + 4q^{13} + 4q^{14} - 4q^{16} - 2q^{17} + 4q^{18} + O(q^{20}) \)  

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

Modular degree:	3360
\( \Gamma_0(N) \)-optimal:	yes
Manin constant:	1

Special L-value

\( L(E,1) \) ≈ \( 0.78682716999922047934734423023914056002 \)

Local data

This elliptic curve is not semistable. There are 2 primes of bad reduction:

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(5\)	\(1\)	\(I_0^{*}\)	Additive	1	2	6	0
\(11\)	\(2\)	\(I_1^{*}\)	Additive	-1	2	7	1

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.4.1

$p$-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	ss	ordinary	add	ordinary	add	ordinary	ordinary	ss	ordinary	ss	ordinary	ordinary	ordinary	ordinary	ordinary
$\lambda$-invariant(s)	?	2	-	0	-	0	0	0,0	2	0,0	0	0	0	0	0
$\mu$-invariant(s)	?	0	-	0	-	0	0	0,0	0	0,0	0	0	0	0	0

An entry ? indicates that the invariants have not yet been computed.

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 and 25.
Its isogeny class 3025g consists of 3 curves linked by isogenies of degrees dividing 25.

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
$2$	\(\Q(\sqrt{-55}) \)	\(\Z/5\Z\)	Not in database
$3$	3.1.44.1	\(\Z/2\Z\)	Not in database
$6$	6.0.21296.1	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$6$	6.0.2662000.1	\(\Z/10\Z\)	Not in database
$8$	8.2.2421502441875.2	\(\Z/3\Z\)	Not in database
$10$	10.0.7368586534375.1	\(\Z/25\Z\)	Not in database
$12$	Deg 12	\(\Z/4\Z\)	Not in database
$12$	12.0.7086244000000.1	\(\Z/2\Z \times \Z/10\Z\)	Not in database
$16$	Deg 16	\(\Z/15\Z\)	Not in database
$20$	20.4.169675210983039290802001953125.1	\(\Z/5\Z\)	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.