Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-693764x-222471088\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-693764xz^2-222471088z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-899117523x-10376913717522\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-480, 271\right) \) | $1.1555301437271328164256813477$ | $\infty$ |
| \( \left(-\frac{1929}{4}, \frac{1925}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-480:271:1]\) | $1.1555301437271328164256813477$ | $\infty$ |
| \([-3858:1925:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-17277, 6804\right) \) | $1.1555301437271328164256813477$ | $\infty$ |
| \( \left(-17358, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-480, 271\right) \), \( \left(-480, 208\right) \), \( \left(1158, 22384\right) \), \( \left(1158, -23543\right) \), \( \left(2940, 150688\right) \), \( \left(2940, -153629\right) \)
\([-480:271:1]\), \([-480:208:1]\), \([1158:22384:1]\), \([1158:-23543:1]\), \([2940:150688:1]\), \([2940:-153629:1]\)
\((-17277,\pm 6804)\), \((41691,\pm 4960116)\), \((105843,\pm 32866236)\)
Invariants
| Conductor: | $N$ | = | \( 11130 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 53$ |
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| Minimal Discriminant: | $\Delta$ | = | $452758954469850$ | = | $2 \cdot 3^{20} \cdot 5^{2} \cdot 7^{2} \cdot 53 $ |
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| j-invariant: | $j$ | = | \( \frac{36928196050908253259449}{452758954469850} \) | = | $2^{-1} \cdot 3^{-20} \cdot 5^{-2} \cdot 7^{-2} \cdot 53^{-1} \cdot 3491^{3} \cdot 9539^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9601584873615317513855222158$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.9601584873615317513855222158$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9875172218259044$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.577013483597003$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.1555301437271328164256813477$ |
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| Real period: | $\Omega$ | ≈ | $0.16542342526534306903144599769$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 80 $ = $ 1\cdot( 2^{2} \cdot 5 )\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.8230350874539298123587556959 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.823035087 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.165423 \cdot 1.155530 \cdot 80}{2^2} \\ & \approx 3.823035087\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 122880 |
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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 semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $20$ | $I_{20}$ | split multiplicative | -1 | 1 | 20 | 20 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $53$ | $1$ | $I_{1}$ | nonsplit 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 |
|---|---|---|---|
| $2$ | 2B | 8.12.0.7 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1272 = 2^{3} \cdot 3 \cdot 53 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 425 & 8 \\ 428 & 33 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1266 & 1267 \end{array}\right),\left(\begin{array}{rr} 1265 & 8 \\ 1264 & 9 \end{array}\right),\left(\begin{array}{rr} 480 & 1121 \\ 787 & 774 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 440 & 3 \\ 1229 & 2 \end{array}\right),\left(\begin{array}{rr} 803 & 796 \\ 842 & 165 \end{array}\right)$.
The torsion field $K:=\Q(E[1272])$ is a degree-$11886870528$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1272\Z)$.
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 | $4$ | \( 53 \) |
| $3$ | split multiplicative | $4$ | \( 3710 = 2 \cdot 5 \cdot 7 \cdot 53 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 742 = 2 \cdot 7 \cdot 53 \) |
| $7$ | split multiplicative | $8$ | \( 1590 = 2 \cdot 3 \cdot 5 \cdot 53 \) |
| $53$ | nonsplit multiplicative | $54$ | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 11130.m
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{106}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-53}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-53})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | 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 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | split | nonsplit | split | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | nonsplit |
| $\lambda$-invariant(s) | 3 | 2 | 1 | 2 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 1 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.