Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-1228x+9104\)
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(homogenize, simplify) |
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\(y^2z=x^3-1228xz^2+9104z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1228x+9104\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 27968 \) | = | $2^{6} \cdot 19 \cdot 23$ |
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| Discriminant: | $\Delta$ | = | $82710102016$ | = | $2^{19} \cdot 19^{3} \cdot 23 $ |
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| j-invariant: | $j$ | = | \( \frac{781229961}{315514} \) | = | $2^{-1} \cdot 3^{3} \cdot 19^{-3} \cdot 23^{-1} \cdot 307^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.79277873121965587518114689429$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.24694203962026208894470128790$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8615074055151933$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.2184413180833618$ | |||
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: | $\Omega$ | ≈ | $0.98086846071020061393497103276$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2^{2}\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.9234738428408024557398841311 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 3.923473843 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.980868 \cdot 1.000000 \cdot 4}{1^2} \\ & \approx 3.923473843\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 27648 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{9}^{*}$ | additive | -1 | 6 | 19 | 1 |
| $19$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $23$ | $1$ | $I_{1}$ | 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 \( 3496 = 2^{3} \cdot 19 \cdot 23 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 553 & 2 \\ 553 & 3 \end{array}\right),\left(\begin{array}{rr} 2623 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 3495 & 0 \end{array}\right),\left(\begin{array}{rr} 3495 & 2 \\ 3494 & 3 \end{array}\right),\left(\begin{array}{rr} 1749 & 2 \\ 1749 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 3041 & 2 \\ 3041 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[3496])$ is a degree-$25262380154880$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3496\Z)$.
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 | $4$ | \( 437 = 19 \cdot 23 \) |
| $3$ | good | $2$ | \( 1472 = 2^{6} \cdot 23 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 1472 = 2^{6} \cdot 23 \) |
| $23$ | nonsplit multiplicative | $24$ | \( 1216 = 2^{6} \cdot 19 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 27968.bz consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 874.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 |
|---|---|---|---|
| $3$ | 3.3.3496.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.6.42728167936.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.10027208982528.4 | \(\Z/3\Z\) | 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 | ss | ord | ord | ord | ord | ord | nonsplit | nonsplit | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0,0 | 0 | 0 | 2 | 0 |
| $\mu$-invariant(s) | - | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.