Elliptic curve with LMFDB label 3528.w4 (Cremona label 3528x4)

• Label: 3528.w4
• Conductor: $3528$
• Discriminant: $-6.326\times 10^{14}$
• j-invariant: \( \frac{11696828}{7203} \)
• CM: no
• Rank: $0$
• Torsion structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\(y^2=x^3+21021x+297038\)

Mordell-Weil group structure

\(\Z/{2}\Z\)

Torsion generators

\( \left(-14, 0\right) \)

Integral points

\( \left(-14, 0\right) \)

Invariants

Conductor:	\( 3528 \)	=	\(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Discriminant:	\(-632599930432512 \)	=	\(-1 \cdot 2^{10} \cdot 3^{7} \cdot 7^{10} \)
j-invariant:	\( \frac{11696828}{7203} \)	=	\(2^{2} \cdot 3^{-1} \cdot 7^{-4} \cdot 11^{3} \cdot 13^{3}\)
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:	\(0.31669141265948768412742381501\)
Tamagawa product:	\( 32 \)  = \( 2\cdot2^{2}\cdot2^{2} \)
Torsion order:	\(2\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form   3528.2.a.w

\( q + 2q^{5} + 2q^{13} + 6q^{17} + 4q^{19} + O(q^{20}) \)

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

Modular degree:	12288
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L(E,1) \) ≈ \( 2.5335313012759014730193905201058797751 \)

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\((j)_{-}\)
\(2\)	\(2\)	\(III^{*}\)	Additive	-1	3	10	0
\(3\)	\(4\)	\(I_1^{*}\)	Additive	-1	2	7	1
\(7\)	\(4\)	\(I_4^{*}\)	Additive	-1	2	10	4

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 X13b.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 4 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 4 & 1 \end{array}\right)$ and has index 12.

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

prime	Image of Galois representation
\(2\)	B

$p$-adic data

$p$-adic regulators

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

Iwasawa invariants

$p$	2	3	7
Reduction type	add	add	add
$\lambda$-invariant(s)	-	-	-
$\mu$-invariant(s)	-	-	-

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

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 \(d=\) 2 and 4.
Its isogeny class 3528.w consists of 3 curves linked by isogenies of degrees dividing 4.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$2$	\(\Q(\sqrt{-3}) \)	\(\Z/2\Z \times \Z/2\Z\)	Not in database
$2$	\(\Q(\sqrt{7}) \)	\(\Z/4\Z\)	Not in database
$2$	\(\Q(\sqrt{-21}) \)	\(\Z/4\Z\)	Not in database
$4$	\(\Q(\sqrt{-3}, \sqrt{7})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	8.0.7169347584.24	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$8$	8.4.114709561344.13	\(\Z/8\Z\)	Not in database
$8$	8.0.796594176.7	\(\Z/2\Z \times \Z/8\Z\)	Not in database
$8$	8.2.21341355420672.10	\(\Z/6\Z\)	Not in database
$16$	Deg 16	\(\Z/4\Z \times \Z/4\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/8\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$16$	Deg 16	\(\Z/12\Z\)	Not in database
$16$	Deg 16	\(\Z/12\Z\)	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.