Elliptic Curve with LMFDB label 17.a3 (Cremona label 17a1)

• Label: 17.a3
• Conductor: 17
• Discriminant: -83521
• j-invariant: \( -\frac{35937}{83521} \)
• CM: no
• Rank: 0
• Torsion Structure: \(\Z/{4}\Z\)

This is a model for the modular curve $X_0(17)$.

Minimal Weierstrass equation

\( y^2 + x y + y = x^{3} - x^{2} - x - 14 \)

Mordell-Weil group structure

\(\Z/{4}\Z\)

Torsion generators

\( \left(7, 13\right) \)

Integral points

\( \left(7, 13\right) \)

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

Invariants

Conductor:	\( 17 \)	=	\(17\)
Discriminant:	\(-83521 \)	=	\(-1 \cdot 17^{4} \)
j-invariant:	\( -\frac{35937}{83521} \)	=	\(-1 \cdot 3^{3} \cdot 11^{3} \cdot 17^{-4}\)
Endomorphism ring:	\(\Z\)		(no Complex Multiplication)
Sato-Tate Group:	$\mathrm{SU}(2)$

BSD invariants

Rank:	\(0\)
Regulator:	\(1\)
Real period:	\(1.54707975355\)
Tamagawa product:	\( 4 \)  = \( 2^{2} \)
Torsion order:	\(4\)
Analytic order of Ш:	\(1\) (exact)

Modular invariants

Modular form 17.2.a.a

\( q - q^{2} - q^{4} - 2q^{5} + 4q^{7} + 3q^{8} - 3q^{9} + 2q^{10} - 2q^{13} - 4q^{14} - q^{16} + q^{17} + 3q^{18} - 4q^{19} + O(q^{20}) \)

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

Modular degree:	1
\( \Gamma_0(N) \)-optimal:	yes
Manin constant:	1

Special L-value

\( L(E,1) \) ≈ \( 0.386769938388 \)

Local data

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord(\(N\))	ord(\(\Delta\))	ord\((j)_{-}\)
\(17\)	\(4\)	\( I_{4} \)	Split multiplicative	-1	1	4	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 X221a.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^4\Z_2)$ generated by $\left(\begin{array}{rr} 9 & 9 \\ 12 & 3 \end{array}\right),\left(\begin{array}{rr} 5 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 10 \\ 4 & 1 \end{array}\right)$ and has index 96.

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	17
Reduction type	ordinary	split
$\lambda$-invariant(s)	0	1
$\mu$-invariant(s)	2	0

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

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 17.a consists of 4 curves linked by isogenies of degrees dividing 4.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base-change curve
2	\(\Q(\sqrt{-1}) \)	\(\Z/2\Z \times \Z/4\Z\)	2.0.4.1-289.2-a1
4	\(\Q(\zeta_{8})\)	\(\Z/4\Z \times \Z/4\Z\)	Not in database
	4.2.1156.1	\(\Z/8\Z\)	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.

Additional information

This is the only elliptic curve over $\Q$ of prime conductor $N$ and discriminant $\Delta = \pm N^4$, and one of only four prime-conductor curves of discriminant $\pm N^e$ with $e > 2$: the others are the modular curves $X_0(N)$ for N=11 [11.a2] and N=19 [19.a2], and the curve [37.b2] (with $e=5,3,3$ respectively -- in each case, as here with $(N,e)=(17,4)$, the exponent $e$ is the numerator of $(N-1)/12$).