Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-397x+10619\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-397xz^2+10619z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-32184x+7837776\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(29, 156\right) \) | $0.11340071794182733544277014414$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([29:156:1]\) | $0.11340071794182733544277014414$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(264, 4212\right) \) | $0.11340071794182733544277014414$ | $\infty$ |
Integral points
\((-19,\pm 108)\), \((-10,\pm 117)\), \((29,\pm 156)\), \((46,\pm 303)\), \((653,\pm 16692)\)
\([-19:\pm 108:1]\), \([-10:\pm 117:1]\), \([29:\pm 156:1]\), \([46:\pm 303:1]\), \([653:\pm 16692:1]\)
\((-19,\pm 108)\), \((-10,\pm 117)\), \((29,\pm 156)\), \((46,\pm 303)\), \((653,\pm 16692)\)
Invariants
| Conductor: | $N$ | = | \( 8736 \) | = | $2^{5} \cdot 3 \cdot 7 \cdot 13$ |
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| Minimal Discriminant: | $\Delta$ | = | $-45921447936$ | = | $-1 \cdot 2^{12} \cdot 3^{6} \cdot 7 \cdot 13^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{1693669888}{11211291} \) | = | $-1 \cdot 2^{9} \cdot 3^{-6} \cdot 7^{-1} \cdot 13^{-3} \cdot 149^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.72925161071069659139507794992$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.036104430150751281977845828462$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9289275215030577$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.535867221853801$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.11340071794182733544277014414$ |
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| Real period: | $\Omega$ | ≈ | $0.97754592093060795464640880293$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 36 $ = $ 2\cdot( 2 \cdot 3 )\cdot1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.9907587331668858942898655244 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.990758733 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.977546 \cdot 0.113401 \cdot 36}{1^2} \\ & \approx 3.990758733\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6912 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | 1 | 5 | 12 | 0 |
| $3$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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 \( 182 = 2 \cdot 7 \cdot 13 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 181 & 2 \\ 180 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 181 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 15 & 3 \end{array}\right),\left(\begin{array}{rr} 157 & 2 \\ 157 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[182])$ is a degree-$158505984$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/182\Z)$.
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 | $2$ | \( 91 = 7 \cdot 13 \) |
| $3$ | split multiplicative | $4$ | \( 224 = 2^{5} \cdot 7 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 1248 = 2^{5} \cdot 3 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 672 = 2^{5} \cdot 3 \cdot 7 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 8736g consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 8736e1, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.364.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.12057136.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.344128684032.19 | \(\Z/3\Z\) | 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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | split | ord | nonsplit | ord | split | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 2 | 5 | 1 | 1 | 2 | 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 |
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.