Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2-1901800x+1009735552\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z-1901800xz^2+1009735552z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-154045827x+735635079954\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(774, 910)$ | $1.9887044727300095004190947432$ | $\infty$ |
| $(784, 0)$ | $0$ | $2$ |
| $(564, 10780)$ | $0$ | $4$ |
Integral points
\( \left(-1592, 0\right) \), \((-906,\pm 44590)\), \((109,\pm 28350)\), \((564,\pm 10780)\), \((774,\pm 910)\), \( \left(784, 0\right) \), \( \left(809, 0\right) \), \((872,\pm 3696)\), \((1054,\pm 13230)\), \((6724,\pm 540540)\)
Invariants
| Conductor: | $N$ | = | \( 18480 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11$ |
|
| Discriminant: | $\Delta$ | = | $325444692101760000$ | = | $2^{10} \cdot 3^{6} \cdot 5^{4} \cdot 7^{8} \cdot 11^{2} $ |
|
| j-invariant: | $j$ | = | \( \frac{742879737792994384804}{317817082130625} \) | = | $2^{2} \cdot 3^{-6} \cdot 5^{-4} \cdot 7^{-8} \cdot 11^{-2} \cdot 13^{3} \cdot 438877^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3207076372016934455682765861$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.7430849867350723543872498182$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.000725366625225$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.597114271169348$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.9887044727300095004190947432$ |
|
| Real period: | $\Omega$ | ≈ | $0.30005403655361753021903907524$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 512 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot2^{3}\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $8$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $4.7737504364389835795794981944 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 4.773750436 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.300054 \cdot 1.988704 \cdot 512}{8^2} \\ & \approx 4.773750436\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 294912 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{2}^{*}$ | additive | 1 | 4 | 10 | 0 |
| $3$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $7$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $11$ | $2$ | $I_{2}$ | 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 |
|---|---|---|
| $2$ | 2Cs | 4.24.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 887 & 2 \\ 1302 & 1315 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 998 \\ 982 & 1299 \end{array}\right),\left(\begin{array}{rr} 1313 & 988 \\ 18 & 335 \end{array}\right),\left(\begin{array}{rr} 1313 & 8 \\ 1312 & 9 \end{array}\right),\left(\begin{array}{rr} 487 & 6 \\ 714 & 1315 \end{array}\right),\left(\begin{array}{rr} 1057 & 8 \\ 268 & 33 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 1316 & 1317 \end{array}\right)$.
The torsion field $K:=\Q(E[1320])$ is a degree-$2433024000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1320\Z)$.
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 | $2$ | \( 1 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \) |
| $5$ | split multiplicative | $6$ | \( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \) |
| $7$ | split multiplicative | $8$ | \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 18480.bl
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 9240.bf2, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(i, \sqrt{66})\) | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{11})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{6})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | 16.0.2359562117249079705600000000.20 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | 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 | nonsplit | split | split | nonsplit | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 7 | 2 | 2 | 1 | 1 | 1 | 1 | 1,1 | 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.