Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-729x-1455\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-729xz^2-1455z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-945459x-53705970\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-20, 85\right) \) | $3.1796929107285966771614082030$ | $\infty$ |
| \( \left(-2, 1\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-20:85:1]\) | $3.1796929107285966771614082030$ | $\infty$ |
| \([-2:1:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-705, 16200\right) \) | $3.1796929107285966771614082030$ | $\infty$ |
| \( \left(-57, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-20, 85\right) \), \( \left(-20, -65\right) \), \( \left(-2, 1\right) \)
\([-20:85:1]\), \([-20:-65:1]\), \([-2:1:1]\)
\((-705,\pm 16200)\), \( \left(-57, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 93138 \) | = | $2 \cdot 3 \cdot 19^{2} \cdot 43$ |
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| Minimal Discriminant: | $\Delta$ | = | $24275674596$ | = | $2^{2} \cdot 3 \cdot 19^{6} \cdot 43 $ |
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| j-invariant: | $j$ | = | \( \frac{912673}{516} \) | = | $2^{-2} \cdot 3^{-1} \cdot 43^{-1} \cdot 97^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.68272318732624802280145314468$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.78949630225697220720306057126$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9086179539168423$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.743507473501503$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.1796929107285966771614082030$ |
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| Real period: | $\Omega$ | ≈ | $0.99031338582826366251356004390$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot1\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.1488924523177634875679361517 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.148892452 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.990313 \cdot 3.179693 \cdot 4}{2^2} \\ & \approx 3.148892452\end{aligned}$$
Modular invariants
Modular form 93138.2.a.g
For more coefficients, see the Downloads section to the right.
| Modular degree: | 82944 |
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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 not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $19$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $43$ | $1$ | $I_{1}$ | split 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 | 8.6.0.4 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1032 = 2^{3} \cdot 3 \cdot 43 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 866 & 1 \\ 599 & 0 \end{array}\right),\left(\begin{array}{rr} 649 & 388 \\ 128 & 903 \end{array}\right),\left(\begin{array}{rr} 517 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 1029 & 4 \\ 1028 & 5 \end{array}\right),\left(\begin{array}{rr} 346 & 1 \\ 343 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[1032])$ is a degree-$20505526272$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1032\Z)$.
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 | $4$ | \( 46569 = 3 \cdot 19^{2} \cdot 43 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 31046 = 2 \cdot 19^{2} \cdot 43 \) |
| $19$ | additive | $182$ | \( 258 = 2 \cdot 3 \cdot 43 \) |
| $43$ | split multiplicative | $44$ | \( 2166 = 2 \cdot 3 \cdot 19^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 93138.g
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 258.g1, its twist by $-19$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{129}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.2980416.2 | \(\Z/4\Z\) | 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 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | nonsplit | ord | ord | ord | ord | ord | add | ord | ord | ord | ord | ord | split | ord |
| $\lambda$-invariant(s) | 2 | 3 | 1 | 1 | 3 | 1 | 1 | - | 1 | 1 | 1 | 1 | 1 | 2 | 1 |
| $\mu$-invariant(s) | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.