Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-5545555275x-158981090551875\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-5545555275xz^2-158981090551875z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-7187039637075x-7417313955193727250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(144514308755327932861596596820752356052098268950/1026671681105019975192076301907927025029529, 44663156056894433810008050692577927669108118766119804965222227551627175/1040273114473505321946589577063270922459163755274545446101206083)$ | $105.51395602242503066857344874$ | $\infty$ |
| $(85990, -42995)$ | $0$ | $2$ |
Integral points
\( \left(85990, -42995\right) \)
Invariants
| Conductor: | $N$ | = | \( 21450 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 11 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-3694472822777343750000000000$ | = | $-1 \cdot 2^{10} \cdot 3^{5} \cdot 5^{18} \cdot 11^{6} \cdot 13^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{1207087636168285491836819264689}{236446260657750000000000} \) | = | $-1 \cdot 2^{-10} \cdot 3^{-5} \cdot 5^{-12} \cdot 11^{-6} \cdot 13^{-3} \cdot 17^{3} \cdot 179^{3} \cdot 1223^{3} \cdot 2861^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.2892672840234681549978560824$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.4845483278064179676974764158$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0481257219134816$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.913258946248787$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $105.51395602242503066857344874$ |
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| Real period: | $\Omega$ | ≈ | $0.0087473840772974329654459096272$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot1\cdot2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.6918843953728891881287808880 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.691884395 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.008747 \cdot 105.513956 \cdot 16}{2^2} \\ & \approx 3.691884395\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 29030400 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $3$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $5$ | $4$ | $I_{12}^{*}$ | additive | 1 | 2 | 18 | 12 |
| $11$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $13$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 17160 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 13 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7801 & 13740 \\ 2190 & 13801 \end{array}\right),\left(\begin{array}{rr} 15591 & 12730 \\ 14750 & 5021 \end{array}\right),\left(\begin{array}{rr} 17149 & 12 \\ 17148 & 13 \end{array}\right),\left(\begin{array}{rr} 2386 & 3435 \\ 765 & 6856 \end{array}\right),\left(\begin{array}{rr} 13166 & 3435 \\ 14845 & 6856 \end{array}\right),\left(\begin{array}{rr} 6863 & 0 \\ 0 & 17159 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 17110 & 17151 \end{array}\right),\left(\begin{array}{rr} 8581 & 13740 \\ 6870 & 13801 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[17160])$ is a degree-$127529385984000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/17160\Z)$.
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 | $4$ | \( 975 = 3 \cdot 5^{2} \cdot 13 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 50 = 2 \cdot 5^{2} \) |
| $5$ | additive | $18$ | \( 143 = 11 \cdot 13 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 21450c
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 4290y3, its twist by $5$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-39}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.7550400.7 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{-39})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.118098000.10 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.352842894617540123098063280058096943355835937500000000.1 | \(\Z/18\Z\) | 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 | nonsplit | nonsplit | add | ord | nonsplit | nonsplit | ss | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 3 | 1 | - | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 1,1 |
| $\mu$-invariant(s) | 0 | 1 | - | 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.