Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-24080217x+45421448121\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-24080217xz^2+45421448121z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-31207961259x+2119276707417126\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{9}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2730, 4179)$ | $2.1288585214454391613180735771$ | $\infty$ |
| $(2940, 4851)$ | $0$ | $9$ |
Integral points
\( \left(-2478, 301035\right) \), \( \left(-2478, -298557\right) \), \( \left(-2058, 294735\right) \), \( \left(-2058, -292677\right) \), \( \left(1176, 136269\right) \), \( \left(1176, -137445\right) \), \( \left(2424, 34779\right) \), \( \left(2424, -37203\right) \), \( \left(2730, 4179\right) \), \( \left(2730, -6909\right) \), \( \left(2940, 4851\right) \), \( \left(2940, -7791\right) \), \( \left(3234, 35427\right) \), \( \left(3234, -38661\right) \), \( \left(4746, 192675\right) \), \( \left(4746, -197421\right) \), \( \left(15582, 1850583\right) \), \( \left(15582, -1866165\right) \)
Invariants
| Conductor: | $N$ | = | \( 128226 \) | = | $2 \cdot 3 \cdot 7 \cdot 43 \cdot 71$ |
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| Discriminant: | $\Delta$ | = | $2295658741816117891584$ | = | $2^{9} \cdot 3^{9} \cdot 7^{9} \cdot 43^{3} \cdot 71 $ |
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| j-invariant: | $j$ | = | \( \frac{1544204814149745316374461713}{2295658741816117891584} \) | = | $2^{-9} \cdot 3^{-9} \cdot 7^{-9} \cdot 17^{3} \cdot 43^{-3} \cdot 71^{-1} \cdot 197^{3} \cdot 345133^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9997031886550702244696601445$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.9997031886550702244696601445$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9792554099204345$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.3227940769512125$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.1288585214454391613180735771$ |
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| Real period: | $\Omega$ | ≈ | $0.14555518375523455607108591137$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2187 $ = $ 3^{2}\cdot3^{2}\cdot3^{2}\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $9$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.3663926185029717346213588869 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.366392619 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.145555 \cdot 2.128859 \cdot 2187}{9^2} \\ & \approx 8.366392619\end{aligned}$$
Modular invariants
Modular form 128226.2.a.s
For more coefficients, see the Downloads section to the right.
| Modular degree: | 16796160 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $3$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $7$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $43$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $71$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $3$ | 3B.1.1 | 9.72.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1538712 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 43 \cdot 71 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 384679 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 769347 & 1538704 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 858826 & 9 \\ 1073511 & 1538704 \end{array}\right),\left(\begin{array}{rr} 1538695 & 18 \\ 1538694 & 19 \end{array}\right),\left(\begin{array}{rr} 1343674 & 9 \\ 21663 & 1538704 \end{array}\right),\left(\begin{array}{rr} 769358 & 1154043 \\ 384633 & 598186 \end{array}\right),\left(\begin{array}{rr} 659458 & 9 \\ 1099071 & 1538704 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right)$.
The torsion field $K:=\Q(E[1538712])$ is a degree-$6989597150768057548800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1538712\Z)$.
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 | $4$ | \( 64113 = 3 \cdot 7 \cdot 43 \cdot 71 \) |
| $3$ | split multiplicative | $4$ | \( 71 \) |
| $7$ | split multiplicative | $8$ | \( 18318 = 2 \cdot 3 \cdot 43 \cdot 71 \) |
| $43$ | split multiplicative | $44$ | \( 2982 = 2 \cdot 3 \cdot 7 \cdot 71 \) |
| $71$ | nonsplit multiplicative | $72$ | \( 1806 = 2 \cdot 3 \cdot 7 \cdot 43 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 128226ba
consists of 3 curves linked by isogenies of
degrees dividing 9.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{9}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.3.512904.1 | \(\Z/18\Z\) | not in database |
| $6$ | 6.6.134929918510539264.1 | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
| $6$ | 6.0.686115387.1 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $9$ | 9.3.17355261557522246455194664529088.1 | \(\Z/27\Z\) | not in database |
| $12$ | deg 12 | \(\Z/36\Z\) | not in database |
| $18$ | 18.0.317429342877822414921955513624188199742591761907712.1 | \(\Z/3\Z \oplus \Z/18\Z\) | 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 | 71 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | split | ord | split | ord | ord | ss | ord | ord | ord | ord | ord | ord | split | ord | nonsplit |
| $\lambda$-invariant(s) | 4 | 4 | 1 | 2 | 1 | 1 | 1,1 | 3 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.