Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-5473x+1251871\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-5473xz^2+1251871z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-443340x+913943952\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{4111}{9}, \frac{262144}{27}\right) \) | $1.6984050693746808917819760719$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([12333:262144:27]\) | $1.6984050693746808917819760719$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(4114, 262144\right) \) | $1.6984050693746808917819760719$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 1216 \) | = | $2^{6} \cdot 19$ |
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| Minimal Discriminant: | $\Delta$ | = | $-668503069687808$ | = | $-1 \cdot 2^{45} \cdot 19 $ |
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| j-invariant: | $j$ | = | \( -\frac{69173457625}{2550136832} \) | = | $-1 \cdot 2^{-27} \cdot 5^{3} \cdot 19^{-1} \cdot 821^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5248903817971615450778101023$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.48516961095724358095196192011$ |
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| $abc$ quality: | $Q$ | ≈ | $1.054621266416818$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.857309765642934$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.6984050693746808917819760719$ |
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| Real period: | $\Omega$ | ≈ | $0.42519312144919294835791211598$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.8886006117302146694971153443 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.888600612 \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 0.425193 \cdot 1.698405 \cdot 4}{1^2} \\ & \approx 2.888600612\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3456 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{35}^{*}$ | additive | -1 | 6 | 45 | 27 |
| $19$ | $1$ | $I_{1}$ | 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 |
|---|---|---|---|
| $3$ | 3B | 27.36.0.1 | $36$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4104 = 2^{3} \cdot 3^{3} \cdot 19 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 937 & 75 \\ 1653 & 2144 \end{array}\right),\left(\begin{array}{rr} 31 & 36 \\ 2386 & 1447 \end{array}\right),\left(\begin{array}{rr} 4076 & 4077 \\ 2583 & 512 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 688 & 9 \\ 791 & 664 \end{array}\right),\left(\begin{array}{rr} 1025 & 4050 \\ 0 & 4103 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 729 & 703 \end{array}\right),\left(\begin{array}{rr} 4051 & 54 \\ 4050 & 55 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[4104])$ is a degree-$45954293760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4104\Z)$.
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 | $4$ | \( 19 \) |
| $19$ | split multiplicative | $20$ | \( 64 = 2^{6} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 1216o
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 38a2, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/3\Z\) | 2.2.24.1-722.1-g2 |
| $3$ | 3.1.152.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.3511808.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.600519168.2 | \(\Z/3\Z\) | not in database |
| $6$ | 6.6.1313335420416.3 | \(\Z/9\Z\) | not in database |
| $6$ | 6.2.19961856.1 | \(\Z/6\Z\) | not in database |
| $12$ | 12.2.119973433931988992.13 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | 12.0.13235395112984641536.3 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.5003429658525894095216446537728.2 | \(\Z/6\Z\) | not in database |
| $18$ | 18.6.52337641454546532557336143519201345142784.2 | \(\Z/18\Z\) | 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 | add | ord | ss | ord | ord | ord | ord | split | ord | ord | ord | ord | ss | ord | ss |
| $\lambda$-invariant(s) | - | 5 | 1,3 | 1 | 1 | 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 | 0 | 0,0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.