Elliptic curve with Cremona label 21a5 (LMFDB label 21.a1)

• Label: 21a5
• Conductor: $21$
• Discriminant: $147$
• j-invariant: \( \frac{53297461115137}{147} \)
• CM: no
• Rank: $0$
• Torsion structure: \(\Z/{2}\Z\)

Minimal Weierstrass equation

\(y^2+xy=x^3-784x-8515\)

Mordell-Weil group structure

$\Z/{2}\Z$

Torsion generators

\( \left(-\frac{65}{4}, \frac{65}{8}\right) \)

Integral points

None

Invariants

Conductor:	\( 21 \)	=	$3 \cdot 7$
Discriminant:	$147 $	=	$3 \cdot 7^{2} $
j-invariant:	\( \frac{53297461115137}{147} \)	=	$3^{-1} \cdot 7^{-2} \cdot 37633^{3}$
Endomorphism ring:	$\Z$
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)
Sato-Tate group:	$\mathrm{SU}(2)$
Faltings height:	$0.074205383390939770758265854672\dots$
Stable Faltings height:	$0.074205383390939770758265854672\dots$

BSD invariants

Analytic rank:	$0$
Regulator:	$1$
Real period:	$0.90223081077698411118680031280\dots$
Tamagawa product:	$ 2 $  = $ 1\cdot2 $
Torsion order:	$2$
Analytic order of Ш:	$1$ (exact)
Special value:	$ L(E,1) $ ≈ $ 0.45111540538849205559340015640 $

Modular invariants

Modular form   21.2.a.a

\( q - q^{2} + q^{3} - q^{4} - 2 q^{5} - q^{6} - q^{7} + 3 q^{8} + q^{9} + 2 q^{10} + 4 q^{11} - q^{12} - 2 q^{13} + q^{14} - 2 q^{15} - q^{16} - 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20}) \)

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

Modular degree:	4
$ \Gamma_0(N) $-optimal:	no
Manin constant:	1

Local data

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

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord($N$)	ord($\Delta$)	ord$(j)_{-}$
$3$	$1$	$I_{1}$	Split multiplicative	-1	1	1	1
$7$	$2$	$I_{2}$	Non-split 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	16.48.0.48

$p$-adic regulators

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

Iwasawa invariants

$p$	2	3	7
Reduction type	ord	split	nonsplit
$\lambda$-invariant(s)	1	1	0
$\mu$-invariant(s)	2	0	0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4 and 8.
Its isogeny class 21a consists of 6 curves linked by isogenies of degrees dividing 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$2$	\(\Q(\sqrt{3}) \)	\(\Z/2\Z \times \Z/2\Z\)	2.2.12.1-147.1-a7
$2$	\(\Q(\sqrt{-3}) \)	\(\Z/4\Z\)	2.0.3.1-147.2-a8
$2$	\(\Q(\sqrt{-1}) \)	\(\Z/4\Z\)	2.0.4.1-441.1-a6
$4$	\(\Q(\zeta_{12})\)	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$4$	4.2.338688.3	\(\Z/2\Z \times \Z/4\Z\)	Not in database
$4$	4.0.432.1	\(\Z/8\Z\)	Not in database
$4$	\(\Q(i, \sqrt{42})\)	\(\Z/8\Z\)	Not in database
$4$	\(\Q(i, \sqrt{14})\)	\(\Z/8\Z\)	Not in database
$8$	8.0.114709561344.25	\(\Z/4\Z \times \Z/4\Z\)	Not in database
$8$	8.0.2985984.1	\(\Z/2\Z \times \Z/8\Z\)	Not in database
$8$	8.0.12745506816.9	\(\Z/2\Z \times \Z/8\Z\)	Not in database
$8$	8.0.7023034368.6	\(\Z/16\Z\)	Not in database
$8$	8.2.425329947.2	\(\Z/6\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/8\Z\)	Not in database
$16$	Deg 16	\(\Z/4\Z \times \Z/8\Z\)	Not in database
$16$	Deg 16	\(\Z/16\Z\)	Not in database
$16$	Deg 16	\(\Z/16\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/16\Z\)	Not in database
$16$	Deg 16	\(\Z/2\Z \times \Z/6\Z\)	Not in database
$16$	Deg 16	\(\Z/12\Z\)	Not in database
$16$	Deg 16	\(\Z/12\Z\)	Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.