Elliptic curve with Cremona label 38a3 (LMFDB label 38.a2)

• Label: 38a3
• Conductor: $38$
• Discriminant: $-152$
• j-invariant: $-\frac{413493625}{152}$
• CM: no
• Rank: $0$
• Torsion structure: $\Z/{3}\Z$

Minimal Weierstrass equation

$y^2+xy+y=x^3-16x+22$

(, )

Mordell-Weil group structure

$\Z/{3}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(2, -1)$	$0$	$3$

Integral points

$\left(2, -1\right)$, $\left(2, -2\right)$

Invariants

Conductor:	$N$	=	$38$	=	$2 \cdot 19$
Discriminant:	$\Delta$	=	$-152$	=	$-1 \cdot 2^{3} \cdot 19$
j-invariant:	$j$	=	$-\frac{413493625}{152}$	=	$-1 \cdot 2^{-3} \cdot 5^{3} \cdot 19^{-1} \cdot 149^{3}$
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:	$h_{\mathrm{Faltings}}$	≈	$-0.61344267771086611044328331685$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-0.61344267771086611044328331685$
$abc$ quality:	$Q$	≈	$0.9328072208159958$
Szpiro ratio:	$\sigma_{m}$	≈	$5.454382877544644$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$
Real period:	$\Omega$	≈	$5.6718966898268956086360245088$
Tamagawa product:	$\prod_{p}c_p$	=	$1$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$3$
Special value:	$L(E,1)$	≈	$0.63021074331409951207066938986$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$    (exact)

BSD formula

$\displaystyle 0.630210743 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 5.671897 \cdot 1.000000 \cdot 1}{3^2} \approx 0.630210743$

Modular invariants

Modular form   38.2.a.a

$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} - q^{8} - 2 q^{9} - 6 q^{11} + q^{12} + 5 q^{13} + q^{14} + q^{16} + 3 q^{17} + 2 q^{18} + q^{19} + O(q^{20})$

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

Modular degree:	18
$\Gamma_0(N)$-optimal:	no
Manin constant:	3

Local data at primes of bad reduction

This elliptic curve is semistable. There are 2 primes $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$1$	$I_{3}$	nonsplit multiplicative	1	1	3	3
$19$	$1$	$I_{1}$	split multiplicative	-1	1	1	1

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
$3$	3B.1.1	27.72.0.1

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $4104 = 2^{3} \cdot 3^{3} \cdot 19$, index $1296$, genus $43$, and generators

$\left(\begin{array}{rr} 3227 & 681 \\ 2323 & 2500 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 1521 & 3592 \end{array}\right),\left(\begin{array}{rr} 31 & 36 \\ 2386 & 1447 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 729 & 703 \end{array}\right),\left(\begin{array}{rr} 3079 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4051 & 54 \\ 4050 & 55 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right),\left(\begin{array}{rr} 484 & 45 \\ 3019 & 1354 \end{array}\right)$.

The torsion field $K:=\Q(E[4104])$ is a degree-$45954293760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4104\Z)$.

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
$2$	nonsplit multiplicative	$4$	$19$
$3$	good	$2$	$19$
$19$	split multiplicative	$20$	$2$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3 and 9.
Its isogeny class 38a consists of 3 curves linked by isogenies of degrees dividing 9.

Twists

This elliptic curve is its own minimal quadratic twist.

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
$3$	3.1.152.1	$\Z/6\Z$	not in database
$3$	3.3.361.1	$\Z/9\Z$	3.3.361.1-152.1-b2
$6$	6.0.3511808.1	$\Z/2\Z \oplus \Z/6\Z$	not in database
$6$	6.0.3518667.2	$\Z/3\Z \oplus \Z/3\Z$	not in database
$6$	6.0.9747.1	$\Z/9\Z$	not in database
$9$	9.3.457662330368.1	$\Z/18\Z$	not in database
$12$	12.2.119973433931988992.10	$\Z/12\Z$	not in database
$18$	18.0.43564677551979246963.1	$\Z/3\Z \oplus \Z/9\Z$	not in database
$18$	18.0.4122698998419163225428590592.1	$\Z/3\Z \oplus \Z/6\Z$	not in database
$18$	18.0.31634955213811766525952.1	$\Z/18\Z$	not in database
$18$	18.0.2037576378429183552150044672.1	$\Z/2\Z \oplus \Z/18\Z$	not in database

We only show fields where the torsion growth is primitive.

Iwasawa invariants

$p$	2	3	19
Reduction type	nonsplit	ord	split
$\lambda$-invariant(s)	1	0	1
$\mu$-invariant(s)	0	0	0

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

$p$-adic regulators

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