Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-235759x-44101900\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-235759xz^2-44101900z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-305543043x-2056701605634\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 31434 \) | = | $2 \cdot 3 \cdot 13^{2} \cdot 31$ |
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| Discriminant: | $\Delta$ | = | $-829124662254954$ | = | $-1 \cdot 2 \cdot 3 \cdot 13^{6} \cdot 31^{5} $ |
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| j-invariant: | $j$ | = | \( -\frac{300238092661681}{171774906} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-1} \cdot 29^{3} \cdot 31^{-5} \cdot 2309^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8088149888238844120659447117$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.52634031009311604403920099092$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0100188200478981$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.705286767226817$ | |||
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: | $\Omega$ | ≈ | $0.10832727202019294611741806269$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.7081818005048236529354515673 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $25$ = $5^2$ (exact) |
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BSD formula
$$\begin{aligned} 2.708181801 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{25 \cdot 0.108327 \cdot 1.000000 \cdot 1}{1^2} \\ & \approx 2.708181801\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 234000 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $31$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
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 |
|---|---|---|
| $5$ | 5B.4.2 | 5.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 48360 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 31 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 14886 & 14885 \\ 1235 & 33476 \end{array}\right),\left(\begin{array}{rr} 48351 & 10 \\ 48350 & 11 \end{array}\right),\left(\begin{array}{rr} 22319 & 0 \\ 0 & 48359 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 937 & 5590 \\ 26000 & 13521 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7086 & 14885 \\ 31915 & 33476 \end{array}\right),\left(\begin{array}{rr} 14886 & 14885 \\ 9295 & 33476 \end{array}\right),\left(\begin{array}{rr} 39066 & 14885 \\ 45565 & 33476 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 48305 & 48241 \end{array}\right)$.
The torsion field $K:=\Q(E[48360])$ is a degree-$17251247849472000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/48360\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 15717 = 3 \cdot 13^{2} \cdot 31 \) |
| $3$ | split multiplicative | $4$ | \( 10478 = 2 \cdot 13^{2} \cdot 31 \) |
| $5$ | good | $2$ | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
| $13$ | additive | $86$ | \( 186 = 2 \cdot 3 \cdot 31 \) |
| $31$ | nonsplit multiplicative | $32$ | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 31434g
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 186b2, its twist by $13$.
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 |
|---|---|---|---|
| $3$ | 3.1.744.1 | \(\Z/2\Z\) | not in database |
| $4$ | 4.0.21125.1 | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.411830784.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $10$ | 10.2.6090133432500000000.2 | \(\Z/5\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/10\Z\) | 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 | nonsplit | split | ord | ord | ord | add | ord | ord | ord | ss | nonsplit | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 1 | 1 | 2 | 0 | 0 | - | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-invariant(s) | 0 | 0 | 1 | 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 $0$.