Elliptic Curve with LMFDB label 448.a1 (Cremona label 448f6)

• Label: 448.a1
• Conductor: 448
• Discriminant: 6576668672
• j-invariant: \( \frac{2251439055699625}{25088} \)
• CM: no
• Rank: 0
• Torsion Structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\( y^2 = x^{3} + x^{2} - 174753 x + 28059871 \)

Mordell-Weil group structure

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

Torsion generators

\( \left(241, 0\right) \)

Integral points

\( \left(241, 0\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

Conductor:	\( 448 \)	=	\(2^{6} \cdot 7\)
Discriminant:	\(6576668672 \)	=	\(2^{27} \cdot 7^{2} \)
j-invariant:	\( \frac{2251439055699625}{25088} \)	=	\(2^{-9} \cdot 5^{3} \cdot 7^{-2} \cdot 11^{3} \cdot 2383^{3}\)
Endomorphism ring:	\(\Z\)		(no Complex Multiplication)
Sato-Tate Group:	$\mathrm{SU}(2)$

BSD invariants

Rank:	\(0\)
Regulator:	\(1\)
Real period:	\(0.937263843983\)
Tamagawa product:	\( 4 \)  = \( 2\cdot2 \)
Torsion order:	\(2\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 448.2.a.a

\( q - 2q^{3} - q^{7} + q^{9} + 4q^{13} + 6q^{17} + 2q^{19} + O(q^{20}) \)

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

Modular degree:	1152
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L(E,1) \) ≈ \( 0.937263843983 \)

Local data

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(2\)	\( I_17^{*} \)	Additive	-1	6	27	9
\(7\)	\(2\)	\( I_{2} \)	Non-split multiplicative	1	1	2	2

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 X17.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 5 \end{array}\right)$ and has index 6.

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
\(3\)	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	ordinary	nonsplit
$\lambda$-invariant(s)	-	2	0
$\mu$-invariant(s)	-	0	0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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, 3, 6, 9 and 18.
Its isogeny class 448.a consists of 6 curves linked by isogenies of degrees dividing 18.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 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	\(\Q(\sqrt{6}) \)	\(\Z/6\Z\)	2.2.24.1-98.1-d6
	\(\Q(\sqrt{2}) \)	\(\Z/2\Z \times \Z/2\Z\)	2.2.8.1-784.1-a10
4	\(\Q(\sqrt{2}, \sqrt{3})\)	\(\Z/2\Z \times \Z/6\Z\)	Not in database
	4.0.392.1	\(\Z/4\Z\)	Not in database
6	6.0.896168448.4	\(\Z/6\Z\)	Not in database
	6.6.24196548096.1	\(\Z/18\Z\)	Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.