Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+x^2-199928x+34174548\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+x^2z-199928xz^2+34174548z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-16194195x+24961828050\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-2, 5880)$ | $0.15274376448510130173473399249$ | $\infty$ |
| $(243, 0)$ | $0$ | $2$ |
Integral points
\((-212,\pm 8190)\), \((-2,\pm 5880)\), \((238,\pm 360)\), \( \left(243, 0\right) \), \((292,\pm 882)\), \((334,\pm 2184)\), \((439,\pm 5586)\), \((1204,\pm 39246)\), \((4702,\pm 321048)\)
Invariants
| Conductor: | $N$ | = | \( 109200 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $4947447278592000$ | = | $2^{13} \cdot 3^{5} \cdot 5^{3} \cdot 7^{6} \cdot 13^{2} $ |
|
| j-invariant: | $j$ | = | \( \frac{1726143065560493}{9662982966} \) | = | $2^{-1} \cdot 3^{-5} \cdot 7^{-6} \cdot 13^{-2} \cdot 139^{3} \cdot 863^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8538086298493405108591685491$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.75830197118087010779174659434$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9654638477781675$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.157487322883726$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.15274376448510130173473399249$ |
|
| Real period: | $\Omega$ | ≈ | $0.43454515805228771208137756154$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 480 $ = $ 2^{2}\cdot5\cdot2\cdot( 2 \cdot 3 )\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $7.9648875935615706945538052524 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 7.964887594 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.434545 \cdot 0.152744 \cdot 480}{2^2} \\ & \approx 7.964887594\end{aligned}$$
Modular invariants
Modular form 109200.2.a.ft
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1013760 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{5}^{*}$ | additive | -1 | 4 | 13 | 1 |
| $3$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $5$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $7$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $13$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7801 & 4 \\ 4682 & 9 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 5459 & 0 \end{array}\right),\left(\begin{array}{rr} 7282 & 1 \\ 7279 & 0 \end{array}\right),\left(\begin{array}{rr} 9556 & 1369 \\ 6825 & 4096 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 4201 & 4 \\ 8402 & 9 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 2188 & 1 \\ 2183 & 0 \end{array}\right),\left(\begin{array}{rr} 10917 & 4 \\ 10916 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$155817722511360$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
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 | $4$ | \( 15 = 3 \cdot 5 \) |
| $3$ | split multiplicative | $4$ | \( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \) |
| $5$ | additive | $10$ | \( 1456 = 2^{4} \cdot 7 \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 15600 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 109200hg
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 13650n2, its twist by $-4$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{30}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.24843000.3 | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | add | split | add | split | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 2 | - | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 |
| $\mu$-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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.