Elliptic curve with Cremona label 320a2 (LMFDB label 320.d2)

• Label: 320a2
• Conductor: $320$
• Discriminant: $409600$
• j-invariant: $\frac{148176}{25}$
• CM: no
• Rank: $0$
• Torsion structure: $\Z/{2}\Z \oplus \Z/{2}\Z$

Minimal Weierstrass equation

$y^2=x^3-28x+48$

(, )

Mordell-Weil group structure

$\Z/{2}\Z \oplus \Z/{2}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(2, 0)$	$0$	$2$
$(4, 0)$	$0$	$2$

Integral points

$\left(-6, 0\right)$, $\left(2, 0\right)$, $\left(4, 0\right)$

Invariants

Conductor:	$N$	=	$320$	=	$2^{6} \cdot 5$
Discriminant:	$\Delta$	=	$409600$	=	$2^{14} \cdot 5^{2}$
j-invariant:	$j$	=	$\frac{148176}{25}$	=	$2^{4} \cdot 3^{3} \cdot 5^{-2} \cdot 7^{3}$
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:	$h_{\mathrm{Faltings}}$	≈	$-0.20221378199618991273847993622$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-1.0108854926494594403919174113$
$abc$ quality:	$Q$	≈	$1.0917548251330267$
Szpiro ratio:	$\sigma_{m}$	≈	$3.7463616499510577$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$
Real period:	$\Omega$	≈	$2.8551639917674604242854261711$
Tamagawa product:	$\prod_{p}c_p$	=	$8$  = $2^{2}\cdot2$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$4$
Special value:	$L(E,1)$	≈	$1.4275819958837302121427130856$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$    (exact)

BSD formula

$\displaystyle 1.427581996 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 2.855164 \cdot 1.000000 \cdot 8}{4^2} \approx 1.427581996$

Modular invariants

Modular form   320.2.a.d

$q - q^{5} + 4 q^{7} - 3 q^{9} + 4 q^{11} + 2 q^{13} + 2 q^{17} + 4 q^{19} + O(q^{20})$

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

Modular degree:	32
$\Gamma_0(N)$-optimal:	no
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	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$4$	$I_{4}^{*}$	additive	-1	6	14	0
$5$	$2$	$I_{2}$	nonsplit multiplicative	1	1	2	2

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
$2$	2Cs	8.48.0.156

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 40.192.3-40.bk.1.2, level $40 = 2^{3} \cdot 5$, index $192$, genus $3$, and generators

$\left(\begin{array}{rr} 39 & 2 \\ 14 & 35 \end{array}\right),\left(\begin{array}{rr} 19 & 32 \\ 0 & 29 \end{array}\right),\left(\begin{array}{rr} 33 & 8 \\ 32 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 15 & 32 \\ 12 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.

The torsion field $K:=\Q(E[40])$ is a degree-$3840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/40\Z)$.

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	$2$	$1$
$5$	nonsplit multiplicative	$6$	$64 = 2^{6}$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 320a consists of 4 curves linked by isogenies of degrees dividing 4.

Twists

The minimal quadratic twist of this elliptic curve is 40a1, its twist by $-8$.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$4$	$\Q(\sqrt{-2}, \sqrt{5})$	$\Z/2\Z \oplus \Z/4\Z$	not in database
$4$	$\Q(\sqrt{2}, \sqrt{5})$	$\Z/2\Z \oplus \Z/4\Z$	not in database
$4$	$\Q(\zeta_{8})$	$\Z/2\Z \oplus \Z/4\Z$	not in database
$8$	8.0.40960000.1	$\Z/4\Z \oplus \Z/4\Z$	not in database
$8$	8.0.6553600.1	$\Z/2\Z \oplus \Z/8\Z$	not in database
$8$	8.2.89579520000.1	$\Z/2\Z \oplus \Z/6\Z$	not in database
$16$	16.0.16777216000000000000.2	$\Z/2\Z \oplus \Z/8\Z$	not in database
$16$	16.8.16777216000000000000.1	$\Z/2\Z \oplus \Z/8\Z$	not in database
$16$	16.0.26843545600000000.2	$\Z/4\Z \oplus \Z/8\Z$	not in database

We only show fields where the torsion growth is primitive.

Iwasawa invariants

$p$	2	5
Reduction type	add	nonsplit
$\lambda$-invariant(s)	-	0
$\mu$-invariant(s)	-	0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ 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 $0$.