Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3+x^2+263775x+869251125\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3+x^2z+263775xz^2+869251125z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+341851725x+40550652708750\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-3205/4, 99305/8)$ | $2.5569215636242516166809741855$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 144150 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 31^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-327510158377344000000$ | = | $-1 \cdot 2^{13} \cdot 3 \cdot 5^{6} \cdot 31^{8} $ |
|
| j-invariant: | $j$ | = | \( \frac{152303}{24576} \) | = | $2^{-13} \cdot 3^{-1} \cdot 17^{3} \cdot 31$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6157895947404019904163625433$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.47825416446674569417012667301$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0430516566438433$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.604004951712538$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.5569215636242516166809741855$ |
|
| Real period: | $\Omega$ | ≈ | $0.13209464546255278039785747189$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 1\cdot1\cdot2^{2}\cdot3 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $4.0530677690700193105174100775 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 4.053067769 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.132095 \cdot 2.556922 \cdot 12}{1^2} \\ & \approx 4.053067769\end{aligned}$$
Modular invariants
Modular form 144150.2.a.n
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6190080 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{13}$ | nonsplit multiplicative | 1 | 1 | 13 | 13 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $31$ | $3$ | $IV^{*}$ | additive | 1 | 2 | 8 | 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 |
|---|---|---|
| $13$ | 13B.4.1 | 13.28.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 48360 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 31 \), index $336$, genus $9$, and generators
$\left(\begin{array}{rr} 14 & 23 \\ 871 & 1431 \end{array}\right),\left(\begin{array}{rr} 19343 & 0 \\ 0 & 48359 \end{array}\right),\left(\begin{array}{rr} 36271 & 43550 \\ 0 & 13951 \end{array}\right),\left(\begin{array}{rr} 16121 & 19370 \\ 25805 & 10011 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 26 & 1 \end{array}\right),\left(\begin{array}{rr} 24181 & 19370 \\ 33865 & 10011 \end{array}\right),\left(\begin{array}{rr} 48335 & 26 \\ 48334 & 27 \end{array}\right),\left(\begin{array}{rr} 31489 & 28990 \\ 39000 & 25489 \end{array}\right),\left(\begin{array}{rr} 1 & 26 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 36271 & 19370 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[48360])$ is a degree-$2464463978496000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/48360\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 72075 = 3 \cdot 5^{2} \cdot 31^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 48050 = 2 \cdot 5^{2} \cdot 31^{2} \) |
| $5$ | additive | $14$ | \( 5766 = 2 \cdot 3 \cdot 31^{2} \) |
| $13$ | good | $2$ | \( 72075 = 3 \cdot 5^{2} \cdot 31^{2} \) |
| $31$ | additive | $362$ | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
13.
Its isogeny class 144150eq
consists of 2 curves linked by isogenies of
degree 13.
Twists
The minimal quadratic twist of this elliptic curve is 5766h1, its twist by $-155$.
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 |
|---|---|---|---|
| $3$ | 3.1.23064.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.12766754304.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.6.115440125.1 | \(\Z/13\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $18$ | 18.6.293993558723508621193728000000000.2 | \(\Z/26\Z\) | 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 | nonsplit | nonsplit | add | ord | ord | ord | ss | ord | ord | ord | add | ord | ord | ss | ord |
| $\lambda$-invariant(s) | 5 | 1 | - | 1 | 1 | 1 | 1,1 | 1 | 3 | 1 | - | 1 | 1 | 1,1 | 1 |
| $\mu$-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.