Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3+x^2-2166188x-1225592755\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3+x^2z-2166188xz^2-1225592755z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-2807379675x-57139144873866\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(2981, 135689\right) \) | $1.7984135525966315518778565184$ | $\infty$ |
| \( \left(-\frac{14035}{16}, \frac{76051}{64}\right) \) | $3.6098944100706796442355459609$ | $\infty$ |
| \( \left(-819, 409\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([2981:135689:1]\) | $1.7984135525966315518778565184$ | $\infty$ |
| \([-56140:76051:64]\) | $3.6098944100706796442355459609$ | $\infty$ |
| \([-819:409:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(107331, 29630880\right) \) | $1.7984135525966315518778565184$ | $\infty$ |
| \( \left(-\frac{126255}{4}, \frac{1296351}{8}\right) \) | $3.6098944100706796442355459609$ | $\infty$ |
| \( \left(-29469, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-827, 1457\right) \), \( \left(-827, -631\right) \), \( \left(-819, 409\right) \), \( \left(2981, 135689\right) \), \( \left(2981, -138671\right) \), \( \left(3555, 188005\right) \), \( \left(3555, -191561\right) \), \( \left(18675, 2534629\right) \), \( \left(18675, -2553305\right) \)
\([-827:1457:1]\), \([-827:-631:1]\), \([-819:409:1]\), \([2981:135689:1]\), \([2981:-138671:1]\), \([3555:188005:1]\), \([3555:-191561:1]\), \([18675:2534629:1]\), \([18675:-2553305:1]\)
\((-29757,\pm 225504)\), \( \left(-29469, 0\right) \), \((107331,\pm 29630880)\), \((127995,\pm 40993128)\), \((672315,\pm 549496872)\)
Invariants
| Conductor: | $N$ | = | \( 93138 \) | = | $2 \cdot 3 \cdot 19^{2} \cdot 43$ |
|
| Minimal Discriminant: | $\Delta$ | = | $2589514563143248128$ | = | $2^{8} \cdot 3^{6} \cdot 19^{9} \cdot 43 $ |
|
| j-invariant: | $j$ | = | \( \frac{23894093340015625}{55042322688} \) | = | $2^{-8} \cdot 3^{-6} \cdot 5^{6} \cdot 19^{-3} \cdot 41^{3} \cdot 43^{-1} \cdot 281^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4141644002456666976709651405$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.94194491066244646766645142456$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9843247587061957$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.840047881152268$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
|
| Mordell-Weil rank: | $r$ | = | $ 2$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.3596870732321163378637729174$ |
|
| Real period: | $\Omega$ | ≈ | $0.12446167209502370045657746876$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{3}\cdot2\cdot2^{2}\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $12.664596594169226287254697037 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 12.664596594 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.124462 \cdot 6.359687 \cdot 64}{2^2} \\ & \approx 12.664596594\end{aligned}$$
Modular invariants
Modular form 93138.2.a.y
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2488320 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $3$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $19$ | $4$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $43$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9804 = 2^{2} \cdot 3 \cdot 19 \cdot 43 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 11 & 2 \\ 9754 & 9795 \end{array}\right),\left(\begin{array}{rr} 6545 & 2 \\ 6540 & 1 \end{array}\right),\left(\begin{array}{rr} 9793 & 12 \\ 9792 & 13 \end{array}\right),\left(\begin{array}{rr} 1538 & 9801 \\ 4671 & 8 \end{array}\right),\left(\begin{array}{rr} 6394 & 3 \\ 4305 & 9796 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 4095 & 820 \\ 4058 & 809 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[9804])$ is a degree-$19723753082880$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9804\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 15523 = 19^{2} \cdot 43 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 31046 = 2 \cdot 19^{2} \cdot 43 \) |
| $19$ | additive | $200$ | \( 258 = 2 \cdot 3 \cdot 43 \) |
| $43$ | split multiplicative | $44$ | \( 2166 = 2 \cdot 3 \cdot 19^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 93138bc
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 4902h1, its twist by $-19$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{817}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-19}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.7353.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-19}, \sqrt{-43})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.10130208217488.3 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.36088866774801.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.19518045849.3 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.9352533356813202937379393471115606096317839270656.1 | \(\Z/18\Z\) | 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 | split | nonsplit | ss | ord | ss | ord | ord | add | ss | ord | ord | ord | ss | split | ss |
| $\lambda$-invariant(s) | 8 | 2 | 6,2 | 2 | 2,4 | 2 | 2 | - | 2,2 | 2 | 2 | 2 | 2,2 | 3 | 2,2 |
| $\mu$-invariant(s) | 0 | 0 | 0,0 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.