Elliptic curve with LMFDB label 324.d2 (Cremona label 324b1)

• Label: 324.d2
• Conductor: $324$
• Discriminant: $-186624$
• j-invariant: $432$
• CM: no
• Rank: $0$
• Torsion structure: $\Z/{3}\Z$

Minimal Weierstrass equation

$y^2=x^3+9x-18$

(, )

Mordell-Weil group structure

$\Z/{3}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(3, 6)$	$0$	$3$

Integral points

$(3,\pm 6)$

Invariants

Conductor:	$N$	=	$324$	=	$2^{2} \cdot 3^{4}$
Discriminant:	$\Delta$	=	$-186624$	=	$-1 \cdot 2^{8} \cdot 3^{6}$
j-invariant:	$j$	=	$432$	=	$2^{4} \cdot 3^{3}$
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:	$h_{\mathrm{Faltings}}$	≈	$-0.30455901738666521857413475594$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-1.3159632820940169372165787887$
$abc$ quality:	$Q$	≈	$0.7737056144690831$
Szpiro ratio:	$\sigma_{m}$	≈	$3.3393436329884603$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$
Real period:	$\Omega$	≈	$1.6541764509399584640857607023$
Tamagawa product:	$\prod_{p}c_p$	=	$9$  = $3\cdot3$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$3$
Special value:	$L(E,1)$	≈	$1.6541764509399584640857607023$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$    (exact)

BSD formula

$\begin{aligned} 1.654176451 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.654176 \cdot 1.000000 \cdot 9}{3^2} \\ & \approx 1.654176451\end{aligned}$

Modular invariants

Modular form   324.2.a.d

$q + 3 q^{5} + 2 q^{7} - 6 q^{11} + 5 q^{13} - 3 q^{17} + 2 q^{19} + O(q^{20})$

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

Modular degree:	36
$\Gamma_0(N)$-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	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$3$	$IV^{*}$	additive	-1	2	8	0
$3$	$3$	$IV$	additive	1	4	6	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
$2$	2G	4.2.0.1
$3$	3B.1.1	3.8.0.1
$5$	5S4	5.5.0.1

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $120 = 2^{3} \cdot 3 \cdot 5$, index $160$, genus $4$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 60 & 1 \end{array}\right),\left(\begin{array}{rr} 61 & 66 \\ 78 & 49 \end{array}\right),\left(\begin{array}{rr} 88 & 69 \\ 27 & 103 \end{array}\right),\left(\begin{array}{rr} 61 & 60 \\ 60 & 61 \end{array}\right),\left(\begin{array}{rr} 109 & 42 \\ 66 & 97 \end{array}\right),\left(\begin{array}{rr} 81 & 100 \\ 10 & 81 \end{array}\right),\left(\begin{array}{rr} 1 & 60 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 61 & 105 \\ 0 & 91 \end{array}\right),\left(\begin{array}{rr} 61 & 60 \\ 90 & 1 \end{array}\right),\left(\begin{array}{rr} 101 & 20 \\ 100 & 21 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 97 \end{array}\right)$.

The torsion field $K:=\Q(E[120])$ is a degree-$221184$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.

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	$2$	$81 = 3^{4}$
$3$	additive	$6$	$2$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 324.d consists of 2 curves linked by isogenies of degree 3.

Twists

The minimal quadratic twist of this elliptic curve is 324.b2, its twist by $-3$.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$3$	3.1.324.1	$\Z/6\Z$	not in database
$6$	6.0.419904.2	$\Z/2\Z \oplus \Z/6\Z$	not in database
$6$	6.0.34992.1	$\Z/3\Z \oplus \Z/3\Z$	not in database
$9$	9.3.669462604992.3	$\Z/9\Z$	not in database
$12$	12.2.1624959306694656.2	$\Z/12\Z$	not in database
$18$	18.0.647677499181836009472.1	$\Z/3\Z \oplus \Z/6\Z$	not in database

We only show fields where the torsion growth is primitive.

Iwasawa invariants

$p$	2	3
Reduction type	add	add
$\lambda$-invariant(s)	-	-
$\mu$-invariant(s)	-	-

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.

$p$-adic regulators

All $p$-adic regulators are identically $1$ since the rank is $0$.