Elliptic curve with LMFDB label 50.a3 (Cremona label 50a1)

• Label: 50.a3
• Conductor: $50$
• Discriminant: $-1250$
• j-invariant: $-\frac{25}{2}$
• CM: no
• Rank: $0$
• Torsion structure: $\Z/{3}\Z$

Minimal Weierstrass equation

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

(, )

Mordell-Weil group structure

$\Z/{3}\Z$

Mordell-Weil generators

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

Integral points

$\left(2, 1\right)$, $\left(2, -4\right)$

Invariants

Conductor:	$N$	=	$50$	=	$2 \cdot 5^{2}$
Discriminant:	$\Delta$	=	$-1250$	=	$-1 \cdot 2 \cdot 5^{4}$
j-invariant:	$j$	=	$-\frac{25}{2}$	=	$-1 \cdot 2^{-1} \cdot 5^{2}$
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:	$h_{\mathrm{Faltings}}$	≈	$-0.72609959614388153275382537999$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-1.2625789002885816576207451577$
$abc$ quality:	$Q$	≈	$1.0904350906962674$
Szpiro ratio:	$\sigma_{m}$	≈	$3.730250695688521$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$
Real period:	$\Omega$	≈	$2.1394949442848902680158039701$
Tamagawa product:	$\prod_{p}c_p$	=	$3$  = $1\cdot3$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$3$
Special value:	$L(E,1)$	≈	$0.71316498142829675600526799003$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$    (exact)

BSD formula

$\begin{aligned} 0.713164981 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.139495 \cdot 1.000000 \cdot 3}{3^2} \\ & \approx 0.713164981\end{aligned}$

Modular invariants

Modular form   50.2.a.a

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

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

Modular degree:	2
$\Gamma_0(N)$-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	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$1$	$I_{1}$	nonsplit multiplicative	1	1	1	1
$5$	$3$	$IV$	additive	-1	2	4	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	8.2.0.1
$3$	3B.1.1	3.8.0.1
$5$	5B.1.3	5.24.0.4

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $120 = 2^{3} \cdot 3 \cdot 5$, index $384$, genus $9$, and generators

$\left(\begin{array}{rr} 46 & 105 \\ 45 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 42 \\ 90 & 61 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 60 & 1 \end{array}\right),\left(\begin{array}{rr} 81 & 10 \\ 40 & 81 \end{array}\right),\left(\begin{array}{rr} 31 & 90 \\ 15 & 91 \end{array}\right),\left(\begin{array}{rr} 1 & 72 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 41 & 80 \\ 40 & 81 \end{array}\right),\left(\begin{array}{rr} 61 & 90 \\ 45 & 91 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 30 & 1 \end{array}\right),\left(\begin{array}{rr} 73 & 90 \\ 0 & 49 \end{array}\right),\left(\begin{array}{rr} 1 & 36 \\ 60 & 1 \end{array}\right)$.

The torsion field $K:=\Q(E[120])$ is a degree-$92160$ 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
$2$	nonsplit multiplicative	$4$	$25 = 5^{2}$
$5$	additive	$14$	$2$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3, 5 and 15.
Its isogeny class 50.a consists of 4 curves linked by isogenies of degrees dividing 15.

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
$3$	3.1.200.1	$\Z/6\Z$	not in database
$4$	$\Q(\zeta_{5})$	$\Z/15\Z$	not in database
$6$	6.0.320000.1	$\Z/2\Z \oplus \Z/6\Z$	not in database
$6$	6.0.270000.1	$\Z/3\Z \oplus \Z/3\Z$	not in database
$9$	9.3.4920750000.1	$\Z/9\Z$	not in database
$10$	10.2.12500000000.1	$\Z/15\Z$	not in database
$12$	12.2.52428800000000.3	$\Z/12\Z$	not in database
$12$	12.0.200000000000.1	$\Z/30\Z$	not in database
$18$	18.0.322486272000000000000.1	$\Z/3\Z \oplus \Z/6\Z$	not in database
$20$	20.0.781250000000000000000.1	$\Z/5\Z \oplus \Z/15\Z$	not in database

We only show fields where the torsion growth is primitive.

Iwasawa invariants

$p$	2	3	5
Reduction type	nonsplit	ord	add
$\lambda$-invariant(s)	2	2	-
$\mu$-invariant(s)	0	0	-

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ 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 $0$.

Additional information

This curve $E$ also has a double cover by Bring's curve $B \subset {\bf P}^4: \sum_{i=1}^5 x_i = \sum_{i=1}^5 x_i^2 = \sum_{i=1}^5 x_i^3 = 0$ of genus $4$. Indeed $B$ has automorphisms by $S_5$ (permuting the projective coordinates $x_i$), and the quotient by a simple transposition $\sigma \in S_5$ is $E$. The $3$-cycles that commute with $\sigma$ act on $B$ by translation by the $3$-torsion points of $E$.