Elliptic curve with Cremona label 24301a1 (LMFDB label 24301.a1)

• Label: 24301a1
• Conductor: $24301$
• Discriminant: $-24301$
• j-invariant: $-\frac{9434056897}{24301}$
• CM: no
• Rank: $3$
• Torsion structure: trivial

Minimal Weierstrass equation

$y^2+xy=x^3-44x+109$

(, )

Mordell-Weil group structure

$\Z \oplus \Z \oplus \Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(4, -3)$	$0.75096097417805300217629236100$	$\infty$
$(3, 1)$	$1.0100907588240367105240619207$	$\infty$
$(7, 9)$	$1.7787472504058659157072455984$	$\infty$

Integral points

$\left(-5, 17\right)$, $\left(-5, -12\right)$, $\left(-4, 17\right)$, $\left(-4, -13\right)$, $\left(3, 1\right)$, $\left(3, -4\right)$, $\left(4, -1\right)$, $\left(4, -3\right)$, $\left(5, 2\right)$, $\left(5, -7\right)$, $\left(7, 9\right)$, $\left(7, -16\right)$, $\left(9, 17\right)$, $\left(9, -26\right)$, $\left(21, 82\right)$, $\left(21, -103\right)$, $\left(28, 131\right)$, $\left(28, -159\right)$, $\left(60, 433\right)$, $\left(60, -493\right)$, $\left(879, 25624\right)$, $\left(879, -26503\right)$

Invariants

Conductor:	$N$	=	$24301$	=	$19 \cdot 1279$
Discriminant:	$\Delta$	=	$-24301$	=	$-1 \cdot 19 \cdot 1279$
j-invariant:	$j$	=	$-\frac{9434056897}{24301}$	=	$-1 \cdot 19^{-1} \cdot 1279^{-1} \cdot 2113^{3}$
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:	$h_{\mathrm{Faltings}}$	≈	$-0.28231562213212539432157180817$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-0.28231562213212539432157180817$
$abc$ quality:	$Q$	≈	$0.7402712607181847$
Szpiro ratio:	$\sigma_{m}$	≈	$2.2748477680883776$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$3$
Mordell-Weil rank:	$r$	=	$3$
Regulator:	$\mathrm{Reg}(E/\Q)$	≈	$1.0885646024251077131442859489$
Real period:	$\Omega$	≈	$3.7962800924739428123580640801$
Tamagawa product:	$\prod_{p}c_p$	=	$1$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$
Special value:	$L^{(3)}(E,1)/3!$	≈	$4.1324961295582487014720929233$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	≈	$1$    (rounded)

BSD formula

$\displaystyle 4.132496130 \approx L^{(3)}(E,1)/3! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 3.796280 \cdot 1.088565 \cdot 1}{1^2} \approx 4.132496130$

Modular invariants

Modular form   24301.2.a.a

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

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

Modular degree:	3584
$\Gamma_0(N)$-optimal:	yes
Manin constant:	1

Local data at primes of bad reduction

This elliptic curve is 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
$1279$	$1$	$I_{1}$	nonsplit multiplicative	1	1	1	1

Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$.

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $97204 = 2^{2} \cdot 19 \cdot 1279$, index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 1 \\ 97203 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 48603 & 2 \\ 48603 & 3 \end{array}\right),\left(\begin{array}{rr} 97203 & 2 \\ 97202 & 3 \end{array}\right),\left(\begin{array}{rr} 7677 & 2 \\ 7677 & 3 \end{array}\right),\left(\begin{array}{rr} 20465 & 2 \\ 20465 & 3 \end{array}\right)$.

The torsion field $K:=\Q(E[97204])$ is a degree-$15802000309636300800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/97204\Z)$.

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	$20$	$1279$
$1279$	nonsplit multiplicative	$1280$	$19$

Isogenies

This curve has no rational isogenies. Its isogeny class 24301a consists of this curve only.

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

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$3$	3.1.97204.1	$\Z/2\Z$	not in database
$6$	6.0.918443426745664.1	$\Z/2\Z \oplus \Z/2\Z$	not in database
$8$	deg 8	$\Z/3\Z$	not in database
$12$	deg 12	$\Z/4\Z$	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	1279
Reduction type	ord	ord	ord	ord	ord	ord	ord	nonsplit	ord	ord	ord	ss	ord	ord	ord	nonsplit
$\lambda$-invariant(s)	5	3	7	3	3	3	3	3	3	3	3	3,3	3	3	3	?
$\mu$-invariant(s)	0	0	0	0	0	0	0	0	0	0	0	0,0	0	0	0	?

An entry ? indicates that the invariants have not yet been computed.

$p$-adic regulators

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.