Elliptic curve with Cremona label 10051b3 (LMFDB label 10051.c1)

• Label: 10051b3
• Conductor: $10051$
• Discriminant: $-2812681891$
• j-invariant: $-\frac{50357871050752}{19}$
• CM: no
• Rank: $2$
• Torsion structure: trivial

Minimal Weierstrass equation

$y^2+y=x^3+x^2-406977x+99796080$

(, )

Mordell-Weil group structure

$\Z \oplus \Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(360, 264)$	$0.33520833616006797055143310157$	$\infty$
$(13213/36, 6769/216)$	$2.8932249383841783705438480152$	$\infty$

Integral points

$\left(224, 4463\right)$, $\left(224, -4464\right)$, $\left(360, 264\right)$, $\left(360, -265\right)$, $\left(368, 0\right)$, $\left(368, -1\right)$, $\left(1050, 28830\right)$, $\left(1050, -28831\right)$

Invariants

Conductor:	$N$	=	$10051$	=	$19 \cdot 23^{2}$
Discriminant:	$\Delta$	=	$-2812681891$	=	$-1 \cdot 19 \cdot 23^{6}$
j-invariant:	$j$	=	$-\frac{50357871050752}{19}$	=	$-1 \cdot 2^{18} \cdot 19^{-1} \cdot 577^{3}$
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:	$h_{\mathrm{Faltings}}$	≈	$1.6011862649558926489234712995$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$0.033439156991317803520094883596$
$abc$ quality:	$Q$	≈	$1.104947099482495$
Szpiro ratio:	$\sigma_{m}$	≈	$5.465090125454083$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$2$
Mordell-Weil rank:	$r$	=	$2$
Regulator:	$\mathrm{Reg}(E/\Q)$	≈	$0.94442984883344241395105604338$
Real period:	$\Omega$	≈	$0.86055825181833166799702650140$
Tamagawa product:	$\prod_{p}c_p$	=	$4$  = $1\cdot2^{2}$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$
Special value:	$L^{(2)}(E,1)/2!$	≈	$3.2509475987086337902439572490$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	≈	$1$    (rounded)

BSD formula

$\displaystyle 3.250947599 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.860558 \cdot 0.944430 \cdot 4}{1^2} \approx 3.250947599$

Modular invariants

Modular form   10051.2.a.c

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

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

Modular degree:	38016
$\Gamma_0(N)$-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	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$19$	$1$	$I_{1}$	nonsplit multiplicative	1	1	1	1
$23$	$4$	$I_0^{*}$	additive	-1	2	6	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
$3$	3B	27.36.0.1

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $23598 = 2 \cdot 3^{3} \cdot 19 \cdot 23$, index $1296$, genus $43$, and generators

$\left(\begin{array}{rr} 70 & 5175 \\ 7567 & 14836 \end{array}\right),\left(\begin{array}{rr} 13249 & 16491 \\ 17043 & 5222 \end{array}\right),\left(\begin{array}{rr} 31 & 36 \\ 17776 & 16837 \end{array}\right),\left(\begin{array}{rr} 3077 & 0 \\ 0 & 23597 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 23545 & 54 \\ 23544 & 55 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 729 & 703 \end{array}\right)$.

The torsion field $K:=\Q(E[23598])$ is a degree-$47959049825280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/23598\Z)$.

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
$19$	nonsplit multiplicative	$20$	$529 = 23^{2}$
$23$	additive	$266$	$19$

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3 and 9.
Its isogeny class 10051b consists of 3 curves linked by isogenies of degrees dividing 9.

Twists

The minimal quadratic twist of this elliptic curve is 19a2, its twist by $-23$.

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
$2$	$\Q(\sqrt{69})$	$\Z/3\Z$	not in database
$3$	3.1.76.1	$\Z/2\Z$	not in database
$6$	6.0.109744.2	$\Z/2\Z \oplus \Z/2\Z$	not in database
$6$	6.0.14270540463.1	$\Z/3\Z$	not in database
$6$	6.6.31209671992581.1	$\Z/9\Z$	not in database
$6$	6.2.1897467984.1	$\Z/6\Z$	not in database
$12$	deg 12	$\Z/4\Z$	not in database
$12$	deg 12	$\Z/3\Z \oplus \Z/3\Z$	not in database
$12$	12.0.7474187781589267170441.1	$\Z/9\Z$	not in database
$12$	deg 12	$\Z/2\Z \oplus \Z/6\Z$	not in database
$18$	18.0.4297228167484676083994283208121413632.1	$\Z/6\Z$	not in database
$18$	18.6.44950524425770151928627097389761944698379063296.1	$\Z/18\Z$	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.

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	ss	ord	ord	ord	ord	ord	ord	nonsplit	add	ord	ord	ord	ord	ord	ord
$\lambda$-invariant(s)	4,7	2	2	2	2	2	2	4	-	2	2	2	2	2	2
$\mu$-invariant(s)	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

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.