Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-1388653315x+35173858341305\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-1388653315xz^2+35173858341305z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1799694696267x+1641098530192378374\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-185373/4, 185369/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 35490 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-363107588998720550537109375000$ | = | $-1 \cdot 2^{3} \cdot 3^{5} \cdot 5^{20} \cdot 7^{4} \cdot 13^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{61354313914516350666047929}{75227254486083984375000} \) | = | $-1 \cdot 2^{-3} \cdot 3^{-5} \cdot 5^{-20} \cdot 7^{-4} \cdot 13^{-2} \cdot 394410409^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.3656202256475128798880464034$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.0831455469167445118613026826$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0440471482173832$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.2449920503197704$ | |||
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: | $\Omega$ | ≈ | $0.027330922209392089411680833577$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 480 $ = $ 3\cdot1\cdot( 2^{2} \cdot 5 )\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.2797106651270507294017000293 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 3.279710665 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.027331 \cdot 1.000000 \cdot 480}{2^2} \\ & \approx 3.279710665\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 77414400 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $3$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $5$ | $20$ | $I_{20}$ | split multiplicative | -1 | 1 | 20 | 20 |
| $7$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $13$ | $2$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
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 |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1964 & 6721 \\ 8983 & 846 \end{array}\right),\left(\begin{array}{rr} 8737 & 10088 \\ 7228 & 7593 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 10913 & 8 \\ 10912 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 4096 & 8723 \\ 6409 & 8958 \end{array}\right),\left(\begin{array}{rr} 7801 & 10088 \\ 3484 & 7593 \end{array}\right),\left(\begin{array}{rr} 6719 & 0 \\ 0 & 10919 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 10914 & 10915 \end{array}\right),\left(\begin{array}{rr} 2419 & 10816 \\ 338 & 4941 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$38954430627840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 507 = 3 \cdot 13^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 5915 = 5 \cdot 7 \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 2366 = 2 \cdot 7 \cdot 13^{2} \) |
| $7$ | split multiplicative | $8$ | \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 35490cv
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2730c4, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{39}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-26}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{-26})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.87329473560576.59 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | 13 |
|---|---|---|---|---|---|
| Reduction type | split | nonsplit | split | split | add |
| $\lambda$-invariant(s) | 4 | 2 | 1 | 1 | - |
| $\mu$-invariant(s) | 1 | 0 | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.
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 $0$.