Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-6071969993x+182109670698057\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-6071969993xz^2+182109670698057z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-97151519883x+11654921773155782\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-89977, 44988)$ | $0$ | $2$ |
| $(45191, -22596)$ | $0$ | $2$ |
Integral points
\( \left(-89977, 44988\right) \), \( \left(45191, -22596\right) \)
Invariants
| Conductor: | $N$ | = | \( 76230 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $867588906485795479643750400$ | = | $2^{20} \cdot 3^{12} \cdot 5^{2} \cdot 7^{4} \cdot 11^{10} $ |
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| j-invariant: | $j$ | = | \( \frac{19170300594578891358373921}{671785075055001600} \) | = | $2^{-20} \cdot 3^{-6} \cdot 5^{-2} \cdot 7^{-4} \cdot 11^{-4} \cdot 267635041^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.2591121822054842727162262402$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.5108584014722441549876318328$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0347968557718765$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.044822916745452$ | |||
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: | $\Omega$ | ≈ | $0.046727666169879726128847325063$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1280 $ = $ ( 2^{2} \cdot 5 )\cdot2^{2}\cdot2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.7382132935903780903077860050 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 3.738213294 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.046728 \cdot 1.000000 \cdot 1280}{4^2} \\ & \approx 3.738213294\end{aligned}$$
Modular invariants
Modular form 76230.2.a.dc
For more coefficients, see the Downloads section to the right.
| Modular degree: | 88473600 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $20$ | $I_{20}$ | split multiplicative | -1 | 1 | 20 | 20 |
| $3$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $11$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
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 |
|---|---|---|
| $2$ | 2Cs | 8.24.0.18 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9240 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 9233 & 8 \\ 9232 & 9 \end{array}\right),\left(\begin{array}{rr} 5281 & 7568 \\ 1804 & 2553 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6719 & 0 \\ 0 & 9239 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 9236 & 9237 \end{array}\right),\left(\begin{array}{rr} 3785 & 1892 \\ 1892 & 4621 \end{array}\right),\left(\begin{array}{rr} 4203 & 638 \\ 3388 & 5325 \end{array}\right),\left(\begin{array}{rr} 1519 & 4202 \\ 6534 & 5875 \end{array}\right),\left(\begin{array}{rr} 8953 & 5038 \\ 858 & 3365 \end{array}\right)$.
The torsion field $K:=\Q(E[9240])$ is a degree-$4904976384000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9240\Z)$.
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 | $4$ | \( 1089 = 3^{2} \cdot 11^{2} \) |
| $3$ | additive | $6$ | \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 10890 = 2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 76230dl
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 2310o2, its twist by $33$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{-33})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{15}, \sqrt{-33})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.189747360000.5 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.7289334581760000.17 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.11662935330816.11 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.42693156000000.17 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 |
|---|---|---|---|---|---|
| Reduction type | split | add | nonsplit | nonsplit | add |
| $\lambda$-invariant(s) | 5 | - | 2 | 0 | - |
| $\mu$-invariant(s) | 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 $0$.