Elliptic curve with LMFDB label 3025.e1 (Cremona label 3025c2)

• Label: 3025.e1
• Conductor: $3025$
• Discriminant: $-4.053\times 10^{14}$
• j-invariant: \( -121 \)
• CM: no
• Rank: $0$
• Torsion structure: trivial

Minimal Weierstrass equation

\(y^2+xy+y=x^3-7626x+1001273\)

Mordell-Weil group structure

trivial

Integral points

None

Invariants

Conductor:	\( 3025 \)	=	\(5^{2} \cdot 11^{2}\)
Discriminant:	\(-405272259390625 \)	=	\(-1 \cdot 5^{6} \cdot 11^{10} \)
j-invariant:	\( -121 \)	=	\(-1 \cdot 11^{2}\)
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:	\(0.45606612681032076055533948800\)
Tamagawa product:	\( 1 \)  = \( 1\cdot1 \)
Torsion order:	\(1\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form   3025.2.a.e

\( q + q^{2} - 2q^{3} - q^{4} - 2q^{6} - 2q^{7} - 3q^{8} + q^{9} + 2q^{12} + q^{13} - 2q^{14} - q^{16} - 5q^{17} + q^{18} - 6q^{19} + O(q^{20}) \)

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

Modular degree:	9240
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L(E,1) \) ≈ \( 0.45606612681032076055533948800234390299 \)

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
\(11\)	\(1\)	\(II^{*}\)	Additive	-1	2	10	0

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X3.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 3 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 0 & 1 \\ 3 & 1 \end{array}\right)$ and has index 2.

The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime	Image of Galois representation
\(11\)	B.10.4

$p$-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	ordinary	ordinary	add	ordinary	add	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ordinary	ss	ordinary
$\lambda$-invariant(s)	?	2	-	0	-	2	0	0	0	0	0	0	0	0,0	0
$\mu$-invariant(s)	?	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 \(d=\) 11.
Its isogeny class 3025.e consists of 2 curves linked by isogenies of degree 11.

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
$3$	3.1.484.1	\(\Z/2\Z\)	Not in database
$6$	6.0.937024.1	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$8$	8.2.2421502441875.2	\(\Z/3\Z\)	Not in database
$10$	10.10.669871503125.1	\(\Z/11\Z\)	Not in database
$12$	Deg 12	\(\Z/4\Z\)	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.