Elliptic curve with Cremona label 1728y1 (LMFDB label 1728.k1)

• Label: 1728y1
• Conductor: $1728$
• Discriminant: $-644972544$
• j-invariant: $-216$
• CM: no
• Rank: $1$
• Torsion structure: trivial

Minimal Weierstrass equation

$y^2=x^3-108x+1296$

(, )

Mordell-Weil group structure

$\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(18, 72)$	$0.17072412902146954095677645967$	$\infty$

Integral points

$(-14,\pm 8)$, $(0,\pm 36)$, $(18,\pm 72)$, $(928,\pm 28268)$

Invariants

Conductor:	$N$	=	$1728$	=	$2^{6} \cdot 3^{3}$
Discriminant:	$\Delta$	=	$-644972544$	=	$-1 \cdot 2^{15} \cdot 3^{9}$
j-invariant:	$j$	=	$-216$	=	$-1 \cdot 2^{3} \cdot 3^{3}$
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z$    (no potential complex multiplication)
Sato-Tate group:	$\mathrm{ST}(E)$	=	$\mathrm{SU}(2)$
Faltings height:	$h_{\mathrm{Faltings}}$	≈	$0.37510416419092823696485610454$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-1.3152890280100856683531179750$
$abc$ quality:	$Q$	≈	$1.2262943855309167$
Szpiro ratio:	$\sigma_{m}$	≈	$3.7368568478295674$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$1$
Mordell-Weil rank:	$r$	=	$1$
Regulator:	$\mathrm{Reg}(E/\Q)$	≈	$0.17072412902146954095677645967$
Real period:	$\Omega$	≈	$1.4076132756683367784845193590$
Tamagawa product:	$\prod_{p}c_p$	=	$12$  = $2^{2}\cdot3$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$
Special value:	$L'(E,1)$	≈	$2.8837626058504140031043896467$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	≈	$1$    (rounded)

BSD formula

$\displaystyle 2.883762606 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.407613 \cdot 0.170724 \cdot 12}{1^2} \approx 2.883762606$

Modular invariants

Modular form   1728.2.a.k

$q - q^{5} + 3 q^{7} - 3 q^{11} - 4 q^{17} - 6 q^{19} + O(q^{20})$

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

Modular degree:	576
$\Gamma_0(N)$-optimal:	yes
Manin constant:	1

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$4$	$I_{5}^{*}$	additive	-1	6	15	0
$3$	$3$	$IV^{*}$	additive	-1	3	9	0

Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$	mod-$\ell$ image	$\ell$-adic image
$3$	3Nn	9.9.0.1

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $72 = 2^{3} \cdot 3^{2}$, index $36$, genus $3$, and generators

$\left(\begin{array}{rr} 53 & 28 \\ 53 & 17 \end{array}\right),\left(\begin{array}{rr} 35 & 54 \\ 27 & 53 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 16 \\ 50 & 51 \end{array}\right),\left(\begin{array}{rr} 5 & 18 \\ 36 & 29 \end{array}\right),\left(\begin{array}{rr} 12 & 11 \\ 49 & 51 \end{array}\right),\left(\begin{array}{rr} 55 & 18 \\ 54 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 17 & 68 \\ 26 & 31 \end{array}\right)$.

The torsion field $K:=\Q(E[72])$ is a degree-$165888$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/72\Z)$.

The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.

$\ell$	Reduction type	Serre weight	Serre conductor
$2$	additive	$4$	$27 = 3^{3}$
$3$	additive	$2$	$32 = 2^{5}$

Isogenies

This curve has no rational isogenies. Its isogeny class 1728y consists of this curve only.

Twists

The minimal quadratic twist of this elliptic curve is 864a1, its twist by $24$.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$3$	3.1.216.1	$\Z/2\Z$	not in database
$6$	6.0.1119744.1	$\Z/2\Z \oplus \Z/2\Z$	not in database
$8$	8.2.573308928.1	$\Z/3\Z$	not in database
$12$	12.2.1925877696823296.2	$\Z/4\Z$	not in database
$16$	16.0.328683126924509184.1	$\Z/3\Z \oplus \Z/3\Z$	not in database

We only show fields where the torsion growth is primitive.

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	add	add	ord	ord	ord	ss	ord	ord	ord	ord	ord	ord	ord	ord	ord
$\lambda$-invariant(s)	-	-	3	1	1	1,1	1	1	1	1	1	1	3	1	1
$\mu$-invariant(s)	-	-	0	0	0	0,0	0	0	0	0	0	0	0	0	0

An entry - indicates that the invariants are not computed because the reduction is additive.

$p$-adic regulators

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.