Elliptic Curve with Cremona label 441c6 (LMFDB label 441.f4)

• Label: 441c6
• Conductor: 441
• Discriminant: -1483273860320763
• j-invariant: \( -\frac{4354703137}{17294403} \)
• CM: no
• Rank: 1
• Torsion Structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\( y^2 + x y = x^{3} - x^{2} - 15003 x - 1979636 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

\(P\)	=	\( \left(296, 4262\right) \)
\(\hat{h}(P)\)	≈	2.63850882313

Torsion generators

\( \left(\frac{659}{4}, -\frac{659}{8}\right) \)

Integral points

\( \left(296, 4262\right) \)

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

Invariants

Conductor:	\( 441 \)	=	\(3^{2} \cdot 7^{2}\)
Discriminant:	\(-1483273860320763 \)	=	\(-1 \cdot 3^{7} \cdot 7^{14} \)
j-invariant:	\( -\frac{4354703137}{17294403} \)	=	\(-1 \cdot 3^{-1} \cdot 7^{-8} \cdot 23^{3} \cdot 71^{3}\)
Endomorphism ring:	\(\Z\)		(no Complex Multiplication)
Sato-Tate Group:	$\mathrm{SU}(2)$

BSD invariants

Rank:	\(1\)
Regulator:	\(2.63850882313\)
Real period:	\(0.196882904034\)
Tamagawa product:	\( 16 \)  = \( 2^{2}\cdot2^{2} \)
Torsion order:	\(2\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 441.2.a.f

\( q + q^{2} - q^{4} - 2q^{5} - 3q^{8} - 2q^{10} - 4q^{11} + 2q^{13} - q^{16} - 6q^{17} - 4q^{19} + O(q^{20}) \)

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

Modular degree:	1536
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	1

Special L-value

\( L'(E,1) \) ≈ \( 2.07790911767 \)

Local data

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(3\)	\(4\)	\( I_1^{*} \)	Additive	-1	2	7	1
\(7\)	\(4\)	\( I_8^{*} \)	Additive	-1	2	14	8

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 X85h.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^4\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 3 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 8 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 7 \end{array}\right)$ and has index 48.

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

\(p\)-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	ordinary	add	ordinary	add	ordinary	ordinary	ordinary	ordinary	ss	ordinary	ss	ordinary	ordinary	ordinary	ss
$\lambda$-invariant(s)	?	-	3	-	1	1	1	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	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 \(d=\) 2, 4 and 8.
Its isogeny class 441c consists of 6 curves linked by isogenies of degrees dividing 8.

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	\(\Q(\sqrt{7}) \)	\(\Z/4\Z\)	2.2.28.1-567.1-c1
	\(\Q(\sqrt{-21}) \)	\(\Z/4\Z\)	Not in database
	\(\Q(\sqrt{-3}) \)	\(\Z/2\Z \times \Z/2\Z\)	2.0.3.1-7203.3-a3
4	\(\Q(\sqrt{-3}, \sqrt{7})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
	\(\Q(\sqrt{2}, \sqrt{-21})\)	\(\Z/8\Z\)	Not in database
	\(\Q(\sqrt{-6}, \sqrt{14})\)	\(\Z/8\Z\)	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.