Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-2289x+98127\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-2289xz^2+98127z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-185436x+72090864\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(159, 1944\right) \) | $0.61630246524202943547538827720$ | $\infty$ |
| \( \left(-3, 324\right) \) | $0.66852189245942422252494973720$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([159:1944:1]\) | $0.61630246524202943547538827720$ | $\infty$ |
| \([-3:324:1]\) | $0.66852189245942422252494973720$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1434, 52488\right) \) | $0.61630246524202943547538827720$ | $\infty$ |
| \( \left(-24, 8748\right) \) | $0.66852189245942422252494973720$ | $\infty$ |
Integral points
\((-57,\pm 216)\), \((-39,\pm 360)\), \((-3,\pm 324)\), \((39,\pm 264)\), \((42,\pm 279)\), \((63,\pm 456)\), \((159,\pm 1944)\), \((402,\pm 8019)\), \((797,\pm 22476)\), \((1119,\pm 37416)\), \((32679,\pm 5907576)\)
\([-57:\pm 216:1]\), \([-39:\pm 360:1]\), \([-3:\pm 324:1]\), \([39:\pm 264:1]\), \([42:\pm 279:1]\), \([63:\pm 456:1]\), \([159:\pm 1944:1]\), \([402:\pm 8019:1]\), \([797:\pm 22476:1]\), \([1119:\pm 37416:1]\), \([32679:\pm 5907576:1]\)
\((-57,\pm 216)\), \((-39,\pm 360)\), \((-3,\pm 324)\), \((39,\pm 264)\), \((42,\pm 279)\), \((63,\pm 456)\), \((159,\pm 1944)\), \((402,\pm 8019)\), \((797,\pm 22476)\), \((1119,\pm 37416)\), \((32679,\pm 5907576)\)
Invariants
| Conductor: | $N$ | = | \( 76224 \) | = | $2^{6} \cdot 3 \cdot 397$ |
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| Minimal Discriminant: | $\Delta$ | = | $-3456730349568$ | = | $-1 \cdot 2^{14} \cdot 3^{12} \cdot 397 $ |
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| j-invariant: | $j$ | = | \( -\frac{80989901008}{210982077} \) | = | $-1 \cdot 2^{4} \cdot 3^{-12} \cdot 17^{3} \cdot 101^{3} \cdot 397^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0935767051703627324984411979$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.28490499451709320484500372287$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8826007413760943$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.24929021184098$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.32334583662221246008543815391$ |
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| Real period: | $\Omega$ | ≈ | $0.69950520622145569457800335330$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2^{2}\cdot( 2^{2} \cdot 3 )\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $10.856740614108952699483624678 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.856740614 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.699505 \cdot 0.323346 \cdot 48}{1^2} \\ & \approx 10.856740614\end{aligned}$$
Modular invariants
Modular form 76224.2.a.x
For more coefficients, see the Downloads section to the right.
| Modular degree: | 129024 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}^{*}$ | additive | 1 | 6 | 14 | 0 |
| $3$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $397$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1588 = 2^{2} \cdot 397 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 1587 & 0 \end{array}\right),\left(\begin{array}{rr} 1587 & 2 \\ 1586 & 3 \end{array}\right),\left(\begin{array}{rr} 795 & 2 \\ 795 & 3 \end{array}\right),\left(\begin{array}{rr} 5 & 2 \\ 5 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[1588])$ is a degree-$1189337707008$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1588\Z)$.
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 | $2$ | \( 397 \) |
| $3$ | split multiplicative | $4$ | \( 25408 = 2^{6} \cdot 397 \) |
| $397$ | nonsplit multiplicative | $398$ | \( 192 = 2^{6} \cdot 3 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 76224.x consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 4764.c1, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.1588.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.4004529472.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | 397 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | split | ord | ord | ord | ord | ss | ss | ord | ord | ord | ord | ord | ord | ord | nonsplit |
| $\lambda$-invariant(s) | - | 3 | 2 | 2 | 2 | 2 | 2,2 | 2,2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | - | 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.