Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-98433x-4393089\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-98433xz^2-4393089z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-7973100x-3178642608\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-289, 0)$ | $0$ | $2$ |
Integral points
\( \left(-289, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 78144 \) | = | $2^{6} \cdot 3 \cdot 11 \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $52825685042921472$ | = | $2^{30} \cdot 3^{3} \cdot 11^{3} \cdot 37^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{402355893390625}{201513996288} \) | = | $2^{-12} \cdot 3^{-3} \cdot 5^{6} \cdot 11^{-3} \cdot 37^{-2} \cdot 2953^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9015946695183994680692035280$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.86187389867848150394335534581$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0125399750027195$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.0922904902461985$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.28396582620198417381219171820$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 72 $ = $ 2^{2}\cdot3\cdot3\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $5.1113848716357151286194509276 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 5.111384872 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.283966 \cdot 1.000000 \cdot 72}{2^2} \\ & \approx 5.111384872\end{aligned}$$
Modular invariants
Modular form 78144.2.a.ct
For more coefficients, see the Downloads section to the right.
| Modular degree: | 663552 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{20}^{*}$ | additive | -1 | 6 | 30 | 12 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $11$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $37$ | $2$ | $I_{2}$ | 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 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9768 = 2^{3} \cdot 3 \cdot 11 \cdot 37 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 4895 & 9756 \\ 4896 & 9755 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4883 & 0 \\ 0 & 9767 \end{array}\right),\left(\begin{array}{rr} 9757 & 9766 \\ 4934 & 4893 \end{array}\right),\left(\begin{array}{rr} 3961 & 12 \\ 4230 & 73 \end{array}\right),\left(\begin{array}{rr} 8134 & 9757 \\ 8151 & 4904 \end{array}\right),\left(\begin{array}{rr} 6909 & 5288 \\ 6946 & 5299 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 9718 & 9759 \end{array}\right),\left(\begin{array}{rr} 6206 & 9765 \\ 6687 & 4892 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 9757 & 12 \\ 9756 & 13 \end{array}\right)$.
The torsion field $K:=\Q(E[9768])$ is a degree-$18472491417600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9768\Z)$.
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 | $2$ | \( 33 = 3 \cdot 11 \) |
| $3$ | split multiplicative | $4$ | \( 2368 = 2^{6} \cdot 37 \) |
| $11$ | split multiplicative | $12$ | \( 7104 = 2^{6} \cdot 3 \cdot 37 \) |
| $37$ | nonsplit multiplicative | $38$ | \( 2112 = 2^{6} \cdot 3 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 78144.ct
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 2442.h4, its twist by $-8$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.2891328.6 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{33})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.25908401664.4 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.9103797810302976.15 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.8359777603584.12 | \(\Z/12\Z\) | 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 |
| $18$ | 18.0.19496358987415459215469565194986328160555948310528.1 | \(\Z/18\Z\) | 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 | 11 | 37 |
|---|---|---|---|---|
| Reduction type | add | split | split | nonsplit |
| $\lambda$-invariant(s) | - | 3 | 1 | 0 |
| $\mu$-invariant(s) | - | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.