Elliptic Curve with LMFDB label 14.a5 (Cremona label 14a4)

• Label: 14.a5
• Conductor: 14
• Discriminant: -28
• j-invariant: \( -\frac{15625}{28} \)
• CM: no
• Rank: 0
• Torsion Structure: \(\Z/{6}\Z\)

This is a model for the modular curve $X_1(14)$.

Minimal Weierstrass equation

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

Mordell-Weil group structure

\(\Z/{6}\Z\)

Torsion generators

\( \left(1, 0\right) \)

Integral points

\( \left(-1, 0\right) \), \( \left(0, 0\right) \), \( \left(1, 0\right) \)

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

Invariants

Conductor:	\( 14 \)	=	\(2 \cdot 7\)
Discriminant:	\(-28 \)	=	\(-1 \cdot 2^{2} \cdot 7 \)
j-invariant:	\( -\frac{15625}{28} \)	=	\(-1 \cdot 2^{-2} \cdot 5^{6} \cdot 7^{-1}\)
Endomorphism ring:	\(\Z\)		(no Complex Multiplication)
Sato-Tate Group:	$\mathrm{SU}(2)$

BSD invariants

Rank:	\(0\)
Regulator:	\(1\)
Real period:	\(5.9440258682\)
Tamagawa product:	\( 2 \)  = \( 2\cdot1 \)
Torsion order:	\(6\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 14.2.a.a

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

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

Modular degree:	3
\( \Gamma_0(N) \)-optimal:	no
Manin constant:	3

Special L-value

\( L(E,1) \) ≈ \( 0.330223659344 \)

Local data

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(2\)	\(2\)	\( I_{2} \)	Non-split multiplicative	1	1	2	2
\(7\)	\(1\)	\( I_{1} \)	Split multiplicative	-1	1	1	1

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 X16.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 2 & 1 \end{array}\right)$ 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.1.1

$p$-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	nonsplit	ordinary	split
$\lambda$-invariant(s)	0	0	1
$\mu$-invariant(s)	0	0	0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 6, 9 and 18.
Its isogeny class 14.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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base-change curve
2	\(\Q(\sqrt{-7}) \)	\(\Z/2\Z \times \Z/6\Z\)	2.0.7.1-28.2-a6
3	\(\Q(\zeta_{7})^+\)	\(\Z/18\Z\)	3.3.49.1-56.1-a5
4	4.2.448.1	\(\Z/12\Z\)	Not in database
6	6.0.21168.1	\(\Z/18\Z\)	Not in database
	6.0.1037232.1	\(\Z/3\Z \times \Z/6\Z\)	Not in database
	\(\Q(\zeta_{7})\)	\(\Z/2\Z \times \Z/18\Z\)	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.

Additional information

This curve $E$ also parametrizes pairs $(R,T)$ where $R$ is a rational rectangle, $T$ is a Pythagorean triangle, and $R,T$ have the same perimeter and the same area. (That is, $E$ is birational with the curve of $(a:b:c:x:y) \in {\bf P}^4$ with $a^2+b^2=c^2$, $a+b+c=2x+2y$, and $ab/2=xy$.) Unfortunately the six rational points on $E$ all yield degenerate solutions. [Noted in passing in Richard Guy's paper "My Favorite Elliptic Curve: A Tale of Two Types of Triangles", Amer. Math. Monthly 102 #9 (Nov. 1995), 771-781.]