Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2+2095182x+9915129972\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z+2095182xz^2+9915129972z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+33522909x+634601841118\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2225997/16, 3316862457/64)$ | $9.8785338586781729511498346579$ | $\infty$ |
| $(37, 99945)$ | $0$ | $3$ |
Integral points
\( \left(37, 99945\right) \), \( \left(37, -99982\right) \)
Invariants
| Conductor: | $N$ | = | \( 126126 \) | = | $2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-43062955620270480838944$ | = | $-1 \cdot 2^{5} \cdot 3^{7} \cdot 7^{4} \cdot 11 \cdot 13^{12} $ |
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| j-invariant: | $j$ | = | \( \frac{581124479497931327}{24602777889339936} \) | = | $2^{-5} \cdot 3^{-1} \cdot 7^{2} \cdot 11^{-1} \cdot 13^{-12} \cdot 97^{3} \cdot 2351^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0228960684230428498713082417$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.8249532077372169024719013754$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0359218426092647$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.070900006226758$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $9.8785338586781729511498346579$ |
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| Real period: | $\Omega$ | ≈ | $0.086457928549902999755499670130$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 72 $ = $ 1\cdot2\cdot3\cdot1\cdot( 2^{2} \cdot 3 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.8326205962511604340436678027 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.832620596 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.086458 \cdot 9.878534 \cdot 72}{3^2} \\ & \approx 6.832620596\end{aligned}$$
Modular invariants
Modular form 126126.2.a.cv
For more coefficients, see the Downloads section to the right.
| Modular degree: | 12441600 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $3$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $7$ | $3$ | $IV$ | additive | 1 | 2 | 4 | 0 |
| $11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
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 |
|---|---|---|
| $3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 264 = 2^{3} \cdot 3 \cdot 11 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 199 & 6 \\ 69 & 19 \end{array}\right),\left(\begin{array}{rr} 133 & 6 \\ 135 & 19 \end{array}\right),\left(\begin{array}{rr} 118 & 141 \\ 263 & 230 \end{array}\right),\left(\begin{array}{rr} 259 & 6 \\ 258 & 7 \end{array}\right),\left(\begin{array}{rr} 145 & 6 \\ 171 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[264])$ is a degree-$60825600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/264\Z)$.
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 | $4$ | \( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \) |
| $3$ | additive | $8$ | \( 1078 = 2 \cdot 7^{2} \cdot 11 \) |
| $5$ | good | $2$ | \( 63063 = 3^{2} \cdot 7^{2} \cdot 11 \cdot 13 \) |
| $7$ | additive | $20$ | \( 2574 = 2 \cdot 3^{2} \cdot 11 \cdot 13 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 11466 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 126126bg
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 42042cy2, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.12936.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.44177785344.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.99636092064432.4 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $9$ | 9.3.1908788887642771532269872.1 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $18$ | 18.0.1960899633722774641095662317519330194947731095552.1 | \(\Z/3\Z \oplus \Z/6\Z\) | 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 | add | ord | add | nonsplit | split | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 6 | - | 1 | - | 1 | 2 | 1 | 3 | 1 | 1 | 1 | 3 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 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.