Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3+10737099x-2249467552\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3+10737099xz^2-2249467552z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+13915280925x-104992903937250\)
|
(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 46550 \) | = | $2 \cdot 5^{2} \cdot 7^{2} \cdot 19$ |
|
| Discriminant: | $\Delta$ | = | $-81408810991747592500000$ | = | $-1 \cdot 2^{5} \cdot 5^{7} \cdot 7^{15} \cdot 19^{3} $ |
|
| j-invariant: | $j$ | = | \( \frac{74469146542554959}{44285662466080} \) | = | $2^{-5} \cdot 5^{-1} \cdot 7^{-9} \cdot 13^{3} \cdot 19^{-3} \cdot 32363^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0854740327458824387731052003$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.3078000020011755989200491620$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0001350474775885$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.599150457735986$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.063250441627792771217341174554$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot2\cdot2^{2}\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L(E,1)$ | ≈ | $0.50600353302234216973872939643 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 0.506003533 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.063250 \cdot 1.000000 \cdot 8}{1^2} \\ & \approx 0.506003533\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 7464960 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $5$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $7$ | $4$ | $I_{9}^{*}$ | additive | -1 | 2 | 15 | 9 |
| $19$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 15960 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 19 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 7981 & 6 \\ 7983 & 19 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 4658 & 11307 \\ 13975 & 7987 \end{array}\right),\left(\begin{array}{rr} 2279 & 15954 \\ 6837 & 15941 \end{array}\right),\left(\begin{array}{rr} 3991 & 6 \\ 11973 & 19 \end{array}\right),\left(\begin{array}{rr} 6383 & 15954 \\ 3189 & 15941 \end{array}\right),\left(\begin{array}{rr} 15955 & 6 \\ 15954 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4201 & 6 \\ 12603 & 19 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[15960])$ is a degree-$549000629452800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/15960\Z)$.
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 | $4$ | \( 23275 = 5^{2} \cdot 7^{2} \cdot 19 \) |
| $3$ | good | $2$ | \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \) |
| $5$ | additive | $18$ | \( 931 = 7^{2} \cdot 19 \) |
| $7$ | additive | $32$ | \( 950 = 2 \cdot 5^{2} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 46550n
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 1330j2, its twist by $-35$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{105}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.5320.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.150568768000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.12502350000.12 | \(\Z/3\Z\) | not in database |
| $6$ | 6.2.26745768000.5 | \(\Z/6\Z\) | 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 |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.212838147703407843959322807752402984062500000000.2 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.1506317435698094676486273024000000000000000.2 | \(\Z/6\Z\) | 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 | ord | add | add | ord | ord | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 5 | 4 | - | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-invariant(s) | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.