Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2-25863496x+47862413296\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z-25863496xz^2+47862413296z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-2094943203x+34885414463202\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2106, 52250)$ | $2.9902096417156120005107619801$ | $\infty$ |
| $(2356, 0)$ | $0$ | $2$ |
| $(3481, 0)$ | $0$ | $2$ |
Integral points
\( \left(-5836, 0\right) \), \((-4142,\pm 289674)\), \((2106,\pm 52250)\), \( \left(2356, 0\right) \), \( \left(3481, 0\right) \), \((39220,\pm 7704576)\)
Invariants
| Conductor: | $N$ | = | \( 18480 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11$ |
|
| Discriminant: | $\Delta$ | = | $117966068508524544000000$ | = | $2^{30} \cdot 3^{4} \cdot 5^{6} \cdot 7^{2} \cdot 11^{6} $ |
|
| j-invariant: | $j$ | = | \( \frac{467116778179943012100169}{28800309694464000000} \) | = | $2^{-18} \cdot 3^{-4} \cdot 5^{-6} \cdot 7^{-2} \cdot 11^{-6} \cdot 31^{3} \cdot 67^{3} \cdot 37357^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1788146103248663803311103732$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.4856674297649210709138782517$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0262644607334475$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.3941155549489395$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.9902096417156120005107619801$ |
|
| Real period: | $\Omega$ | ≈ | $0.10317529053717482901187155072$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 192 $ = $ 2^{2}\cdot2\cdot2\cdot2\cdot( 2 \cdot 3 ) $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $3.7021889826128366241409390063 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 3.702188983 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.103175 \cdot 2.990210 \cdot 192}{4^2} \\ & \approx 3.702188983\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1990656 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{22}^{*}$ | additive | -1 | 4 | 30 | 18 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $5$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $11$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
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 | 8.12.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9240 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 2521 & 12 \\ 5886 & 73 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 9224 & 9233 \end{array}\right),\left(\begin{array}{rr} 4621 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6929 & 9234 \\ 0 & 9239 \end{array}\right),\left(\begin{array}{rr} 5281 & 12 \\ 3966 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 9229 & 12 \\ 9228 & 13 \end{array}\right),\left(\begin{array}{rr} 6161 & 12 \\ 1540 & 1 \end{array}\right),\left(\begin{array}{rr} 7399 & 6 \\ 5538 & 9235 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[9240])$ is a degree-$2452488192000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9240\Z)$.
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 | $4$ | \( 112 = 2^{4} \cdot 7 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
| $11$ | split 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, 3 and 6.
Its isogeny class 18480bn
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 2310g6, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-5})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{77})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-77})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $6$ | 6.0.9073705536.10 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.186606965293056.183 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.3317760000.6 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.8.455583411360000.11 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $18$ | 18.6.715213020642926602616574456530904768000000000000.3 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | nonsplit | nonsplit | split | ord | ord | ord | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 1 | 3 | 1 | 2 | 1 | 1 | 3 | 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 | 0,0 |
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.