Elliptic curve with LMFDB label 6350400.vk1

• Label: 6350400.vk1
• Conductor: $6350400$
• Discriminant: $-6.947\times 10^{15}$
• j-invariant: $-576$
• CM: no
• Rank: $0$
• Torsion structure: trivial

Minimal Weierstrass equation

$y^2=x^3-33075x+4630500$

(, )

Mordell-Weil group structure

trivial

Invariants

Conductor:	$N$	=	$6350400$	=	$2^{6} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}$
Discriminant:	$\Delta$	=	$-6947055801000000$	=	$-1 \cdot 2^{6} \cdot 3^{10} \cdot 5^{6} \cdot 7^{6}$
j-invariant:	$j$	=	$-576$	=	$-1 \cdot 2^{6} \cdot 3^{2}$
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:	$h_{\mathrm{Faltings}}$	≈	$1.7302396102503979052448939106$
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-1.3095182513310396654794825526$
$abc$ quality:	$Q$	≈	$0.7737056144690831$
Szpiro ratio:	$\sigma_{m}$	≈	$2.8229964439325848$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$
Mordell-Weil rank:	$r$	=	$0$
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$
Real period:	$\Omega$	≈	$0.37580497787157000202397108755$
Tamagawa product:	$\prod_{p}c_p$	=	$6$  = $1\cdot3\cdot1\cdot2$
Torsion order:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$
Special value:	$L(E,1)$	≈	$2.2548298672294200121438265253$
Analytic order of Ш:	Ш${}_{\mathrm{an}}$	=	$1$    (exact)

BSD formula

$\begin{aligned} 2.254829867 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.375805 \cdot 1.000000 \cdot 6}{1^2} \\ & \approx 2.254829867\end{aligned}$

Modular invariants

Modular form 6350400.2.a.vk

$q - 2 q^{11} - q^{13} - 3 q^{17} - 2 q^{19} + O(q^{20})$

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

Modular degree:

29030400

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	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$1$	$II$	additive	1	6	6	0
$3$	$3$	$IV^{*}$	additive	1	4	10	0
$5$	$1$	$I_0^{*}$	additive	1	2	6	0
$7$	$2$	$I_0^{*}$	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
$2$	2G	4.2.0.1

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $420 = 2^{2} \cdot 3 \cdot 5 \cdot 7$, index $4$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 139 & 0 \\ 0 & 419 \end{array}\right),\left(\begin{array}{rr} 419 & 315 \\ 210 & 419 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2 & 3 \\ 415 & 413 \end{array}\right),\left(\begin{array}{rr} 83 & 0 \\ 0 & 419 \end{array}\right),\left(\begin{array}{rr} 179 & 0 \\ 0 & 419 \end{array}\right),\left(\begin{array}{rr} 417 & 4 \\ 416 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 419 & 0 \\ 210 & 419 \end{array}\right)$.

The torsion field $K:=\Q(E[420])$ is a degree-$1114767360$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/420\Z)$.

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	$2$	$99225 = 3^{4} \cdot 5^{2} \cdot 7^{2}$
$3$	additive	$4$	$1120 = 2^{5} \cdot 5 \cdot 7$
$5$	additive	$14$	$254016 = 2^{6} \cdot 3^{4} \cdot 7^{2}$
$7$	additive	$26$	$129600 = 2^{6} \cdot 3^{4} \cdot 5^{2}$

Isogenies

This curve has no rational isogenies. Its isogeny class 6350400.vk consists of this curve only.

Twists

The minimal quadratic twist of this elliptic curve is 2592.c1, its twist by $-280$.

Iwasawa invariants

No Iwasawa invariant data is available for this curve.

$p$-adic regulators

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