Elliptic curve with LMFDB label 445280.a1 (Cremona label 445280a1)

• Label: 445280.a1
• Conductor: $445280$
• Discriminant: $-693468385280$
• j-invariant: $-\frac{93386304}{1399205}$
• CM: no
• Rank: $2$
• Torsion structure: trivial

Minimal Weierstrass equation

$y^2=x^3-748x-40832$

(, )

Mordell-Weil group structure

$\Z \oplus \Z$

Mordell-Weil generators

$P$	$\hat{h}(P)$	Order
$(48, 184)$	$1.2438348222381517538859982612$	$\infty$
$(416, 8464)$	$2.2563834365157533078658734823$	$\infty$

Integral points

$(48,\pm 184)$, $(416,\pm 8464)$, $(1632,\pm 65920)$

Invariants

Conductor:	$N$	=	$445280$	=	$2^{5} \cdot 5 \cdot 11^{2} \cdot 23$
Discriminant:	$\Delta$	=	$-693468385280$	=	$-1 \cdot 2^{12} \cdot 5 \cdot 11^{2} \cdot 23^{4}$
j-invariant:	$j$	=	$-\frac{93386304}{1399205}$	=	$-1 \cdot 2^{6} \cdot 3^{3} \cdot 5^{-1} \cdot 11 \cdot 17^{3} \cdot 23^{-4}$
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:	$h_{\mathrm{Faltings}}$	≈	$0.95348961137077334040482293767$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-0.13930678132223372635606644678$
$abc$ quality:	$Q$	≈	$0.9375105098561078$
Szpiro ratio:	$\sigma_{m}$	≈	$2.6723329848697484$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$2$
Mordell-Weil rank:	$r$	=	$2$
Regulator:	$\mathrm{Reg}(E/\Q)$	≈	$2.7901000604027614167436217921$
Real period:	$\Omega$	≈	$0.38796271241190227927118752475$
Tamagawa product:	$\prod_{p}c_p$	=	$8$  = $2\cdot1\cdot1\cdot2^{2}$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$
Special value:	$L^{(2)}(E,1)/2!$	≈	$8.6596382986757416464271450703$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	≈	$1$    (rounded)

BSD formula

$\begin{aligned} 8.659638299 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.387963 \cdot 2.790100 \cdot 8}{1^2} \\ & \approx 8.659638299\end{aligned}$

Modular invariants

Modular form 445280.2.a.a

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

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

Modular degree:	1277952
$\Gamma_0(N)$-optimal:	yes
Manin constant:	1

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$2$	$I_{3}^{*}$	additive	-1	5	12	0
$5$	$1$	$I_{1}$	nonsplit multiplicative	1	1	1	1
$11$	$1$	$II$	additive	-1	2	2	0
$23$	$4$	$I_{4}$	split multiplicative	-1	1	4	4

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 label 20.2.0.a.1, level $20 = 2^{2} \cdot 5$, index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 19 & 0 \end{array}\right),\left(\begin{array}{rr} 17 & 2 \\ 17 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 11 & 3 \end{array}\right),\left(\begin{array}{rr} 19 & 2 \\ 18 & 3 \end{array}\right)$.

The torsion field $K:=\Q(E[20])$ is a degree-$23040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/20\Z)$.

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	$2$	$605 = 5 \cdot 11^{2}$
$5$	nonsplit multiplicative	$6$	$89056 = 2^{5} \cdot 11^{2} \cdot 23$
$11$	additive	$32$	$3680 = 2^{5} \cdot 5 \cdot 23$
$23$	split multiplicative	$24$	$19360 = 2^{5} \cdot 5 \cdot 11^{2}$

Isogenies

This curve has no rational isogenies. Its isogeny class 445280.a consists of this curve only.

Twists

This elliptic curve is its own minimal quadratic twist.

Iwasawa invariants

No Iwasawa invariant data is available for this curve.

$p$-adic regulators

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