Elliptic curve with LMFDB label 208.d1 (Cremona label 208d2)

• Label: 208.d1
• Conductor: $208$
• Discriminant: $-514035851264$
• j-invariant: $-\frac{1064019559329}{125497034}$
• CM: no
• Rank: $0$
• Torsion structure: trivial

Minimal Weierstrass equation

$y^2=x^3-3403x+83834$

(, )

Mordell-Weil group structure

trivial

Invariants

Conductor:	$N$	=	$208$	=	$2^{4} \cdot 13$
Discriminant:	$\Delta$	=	$-514035851264$	=	$-1 \cdot 2^{13} \cdot 13^{7}$
j-invariant:	$j$	=	$-\frac{1064019559329}{125497034}$	=	$-1 \cdot 2^{-1} \cdot 3^{3} \cdot 13^{-7} \cdot 41^{3} \cdot 83^{3}$
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:	$h_{\mathrm{Faltings}}$	≈	$0.98294170666332987461323919148$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$0.28979452610338456519600707002$
$abc$ quality:	$Q$	≈	$1.0626891964834324$
Szpiro ratio:	$\sigma_{m}$	≈	$6.781466073184546$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$
Real period:	$\Omega$	≈	$0.90202859669078213679475868301$
Tamagawa product:	$\prod_{p}c_p$	=	$2$  = $2\cdot1$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$
Special value:	$L(E,1)$	≈	$1.8040571933815642735895173660$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$    (exact)

BSD formula

$\begin{aligned} 1.804057193 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.902029 \cdot 1.000000 \cdot 2}{1^2} \\ & \approx 1.804057193\end{aligned}$

Modular invariants

Modular form   208.2.a.d

$q + 3 q^{3} - q^{5} - q^{7} + 6 q^{9} + 2 q^{11} - q^{13} - 3 q^{15} - 3 q^{17} - 6 q^{19} + O(q^{20})$

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

Modular degree:	336
$\Gamma_0(N)$-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	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$2$	$I_{5}^{*}$	additive	-1	4	13	1
$13$	$1$	$I_{7}$	nonsplit multiplicative	1	1	7	7

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
$7$	7B.6.3	7.24.0.2

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $728 = 2^{3} \cdot 7 \cdot 13$, index $96$, genus $2$, and generators

$\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 185 & 528 \\ 350 & 561 \end{array}\right),\left(\begin{array}{rr} 720 & 721 \\ 553 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 120 & 7 \\ 441 & 722 \end{array}\right),\left(\begin{array}{rr} 8 & 7 \\ 357 & 722 \end{array}\right),\left(\begin{array}{rr} 715 & 14 \\ 714 & 15 \end{array}\right)$.

The torsion field $K:=\Q(E[728])$ is a degree-$845365248$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/728\Z)$.

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	$4$	$13$
$7$	good	$2$	$16 = 2^{4}$
$13$	nonsplit multiplicative	$14$	$16 = 2^{4}$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 7.
Its isogeny class 208.d consists of 2 curves linked by isogenies of degree 7.

Twists

The minimal quadratic twist of this elliptic curve is 26.b1, its twist by $-4$.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$3$	3.1.104.1	$\Z/2\Z$	not in database
$6$	6.0.1124864.1	$\Z/2\Z \oplus \Z/2\Z$	not in database
$6$	$\Q(\zeta_{28})^+$	$\Z/7\Z$	not in database
$8$	8.2.63962016768.6	$\Z/3\Z$	not in database
$12$	12.2.8421963387109376.10	$\Z/4\Z$	not in database
$14$	14.0.711168436835713024.1	$\Z/7\Z$	not in database
$18$	18.6.24605408747562862612093861888.1	$\Z/14\Z$	not in database

We only show fields where the torsion growth is primitive.

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	add	ss	ord	ord	ord	nonsplit	ord	ord	ord	ord	ord	ord	ss	ord	ord
$\lambda$-invariant(s)	-	0,0	0	0	0	0	0	0	0	0	0	0	0,0	0	0
$\mu$-invariant(s)	-	0,0	0	0	0	0	0	0	0	0	0	0	0,0	0	0

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 $0$.