Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-2604x-101936\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-2604xz^2-101936z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-2604x-101936\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(138, 1472)$ | $2.5508350649730223698166829146$ | $\infty$ |
Integral points
\((138,\pm 1472)\)
Invariants
| Conductor: | $N$ | = | \( 7488 \) | = | $2^{6} \cdot 3^{2} \cdot 13$ |
|
| Discriminant: | $\Delta$ | = | $-3358825906176$ | = | $-1 \cdot 2^{21} \cdot 3^{6} \cdot 13^{3} $ |
|
| j-invariant: | $j$ | = | \( -\frac{10218313}{17576} \) | = | $-1 \cdot 2^{-3} \cdot 7^{3} \cdot 13^{-3} \cdot 31^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0941071969732946198990586929$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.49491971820067818992441210775$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.947172997962475$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.101234455945386$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.5508350649730223698166829146$ |
|
| Real period: | $\Omega$ | ≈ | $0.31572492973055807403961073135$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2^{2}\cdot1\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $3.2214488865714041075407219327 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 3.221448887 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.315725 \cdot 2.550835 \cdot 4}{1^2} \\ & \approx 3.221448887\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 11520 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{11}^{*}$ | additive | -1 | 6 | 21 | 3 |
| $3$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $1$ | $I_{3}$ | 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 |
|---|---|---|
| $3$ | 3Cs | 3.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 82 \end{array}\right),\left(\begin{array}{rr} 223 & 450 \\ 630 & 395 \end{array}\right),\left(\begin{array}{rr} 919 & 18 \\ 918 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 926 & 927 \\ 477 & 8 \end{array}\right),\left(\begin{array}{rr} 580 & 9 \\ 567 & 916 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 458 & 927 \\ 711 & 8 \end{array}\right)$.
The torsion field $K:=\Q(E[936])$ is a degree-$1086898176$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/936\Z)$.
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 | $4$ | \( 117 = 3^{2} \cdot 13 \) |
| $3$ | additive | $2$ | \( 64 = 2^{6} \) |
| $13$ | nonsplit multiplicative | $14$ | \( 576 = 2^{6} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 7488.h
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 26.a2, its twist by $24$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/3\Z\) | 2.2.24.1-338.1-g2 |
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/3\Z\) | 2.0.8.1-27378.3-c2 |
| $3$ | 3.1.104.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-3})\) | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.1124864.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.2336256.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.86528.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.5458092097536.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.922417564483584.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.1265319018496.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.32709692374219991845752331567104.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.508701135803869313185140260531601408.2 | \(\Z/9\Z\) | 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 | add | ord | ord | ord | nonsplit | ord | ord | ss | ord | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | - | - | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1,1 | 3 | 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.