Elliptic curve with Cremona label 20a1 (LMFDB label 20.a4)

• Label: 20a1
• Conductor: $20$
• Discriminant: $-6400$
• j-invariant: $\frac{21296}{25}$
• CM: no
• Rank: $0$
• Torsion structure: $\Z/{6}\Z$

This is a model for the modular curve $X_0(20)$.

Minimal Weierstrass equation

$y^2=x^3+x^2+4x+4$

(, )

Mordell-Weil group structure

$\Z/{6}\Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(4, 10)$	$0$	$6$

Integral points

$\left(-1, 0\right)$, $(0,\pm 2)$, $(4,\pm 10)$

Invariants

Conductor:	$N$	=	$20$	=	$2^{2} \cdot 5$
Discriminant:	$\Delta$	=	$-6400$	=	$-1 \cdot 2^{8} \cdot 5^{2}$
j-invariant:	$j$	=	$\frac{21296}{25}$	=	$2^{4} \cdot 5^{-2} \cdot 11^{3}$
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:	$h_{\mathrm{Faltings}}$	≈	$-0.58337650523554731646541001985$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-1.0454746256088441894102314342$
$abc$ quality:	$Q$	≈	$0.8396384887826309$
Szpiro ratio:	$\sigma_{m}$	≈	$5.18724658047375$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$
Real period:	$\Omega$	≈	$2.8243751419591137994837895490$
Tamagawa product:	$\prod_{p}c_p$	=	$6$  = $3\cdot2$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$6$
Special value:	$L(E,1)$	≈	$0.47072919032651896658063159151$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$    (exact)

BSD formula

$\displaystyle 0.470729190 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 2.824375 \cdot 1.000000 \cdot 6}{6^2} \approx 0.470729190$

Modular invariants

Modular form   20.2.a.a

$q - 2 q^{3} - q^{5} + 2 q^{7} + q^{9} + 2 q^{13} + 2 q^{15} - 6 q^{17} - 4 q^{19} + O(q^{20})$

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

Modular degree:	1
$\Gamma_0(N)$-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
$5$	$2$	$I_{2}$	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
$2$	2B	8.12.0.37
$3$	3B.1.1	3.8.0.1

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

$\left(\begin{array}{rr} 97 & 24 \\ 96 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 99 & 4 \\ 71 & 65 \end{array}\right),\left(\begin{array}{rr} 13 & 24 \\ 12 & 13 \end{array}\right),\left(\begin{array}{rr} 15 & 16 \\ 74 & 95 \end{array}\right),\left(\begin{array}{rr} 61 & 24 \\ 6 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 22 \\ 14 & 83 \end{array}\right),\left(\begin{array}{rr} 99 & 16 \\ 46 & 3 \end{array}\right)$.

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
$2$	additive	$2$	$1$
$5$	nonsplit multiplicative	$6$	$4 = 2^{2}$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 20a consists of 4 curves linked by isogenies of degrees dividing 6.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$2$	$\Q(\sqrt{-1})$	$\Z/2\Z \oplus \Z/6\Z$	2.0.4.1-100.2-a6
$4$	4.2.400.1	$\Z/12\Z$	not in database
$6$	6.0.270000.1	$\Z/3\Z \oplus \Z/6\Z$	not in database
$8$	8.0.2560000.1	$\Z/2\Z \oplus \Z/12\Z$	not in database
$8$	8.0.6553600.1	$\Z/2\Z \oplus \Z/12\Z$	not in database
$9$	9.3.787320000.1	$\Z/18\Z$	not in database
$12$	12.0.18662400000000.1	$\Z/6\Z \oplus \Z/6\Z$	not in database
$16$	16.0.26843545600000000.2	$\Z/4\Z \oplus \Z/12\Z$	not in database
$16$	16.4.16777216000000000000.4	$\Z/24\Z$	not in database
$18$	18.0.2538998916710400000000.1	$\Z/2\Z \oplus \Z/18\Z$	not in database

We only show fields where the torsion growth is primitive.

Iwasawa invariants

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

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 $0$.