Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-37x+111\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-37xz^2+111z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-3024x+89964\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-5, 14)$ | $0.62237473177427800568118245345$ | $\infty$ |
| $(2, 7)$ | $0.88792113912519224266859915162$ | $\infty$ |
Integral points
\((-5,\pm 14)\), \((2,\pm 7)\), \((3,\pm 6)\), \((10,\pm 29)\), \((11,\pm 34)\), \((163,\pm 2086)\), \((3947,\pm 248002)\)
Invariants
| Conductor: | $N$ | = | \( 29008 \) | = | $2^{4} \cdot 7^{2} \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $-3248896$ | = | $-1 \cdot 2^{8} \cdot 7^{3} \cdot 37 $ |
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| j-invariant: | $j$ | = | \( -\frac{65536}{37} \) | = | $-1 \cdot 2^{16} \cdot 37^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.047354892419117646793167494272$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.99593055005624284601432709444$ |
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| $abc$ quality: | $Q$ | ≈ | $0.7476414959038227$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.2533713976363865$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.51907869682071472668740943863$ |
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| Real period: | $\Omega$ | ≈ | $2.3362760173948461830739000739$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $4.8508444420912248276525517062 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.850844442 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.336276 \cdot 0.519079 \cdot 4}{1^2} \\ & \approx 4.850844442\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3840 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | -1 | 4 | 8 | 0 |
| $7$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $37$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 518 = 2 \cdot 7 \cdot 37 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 517 & 2 \\ 516 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 517 & 0 \end{array}\right),\left(\begin{array}{rr} 297 & 2 \\ 297 & 3 \end{array}\right),\left(\begin{array}{rr} 113 & 2 \\ 113 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[518])$ is a degree-$11020520448$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/518\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 259 = 7 \cdot 37 \) |
| $7$ | additive | $20$ | \( 592 = 2^{4} \cdot 37 \) |
| $37$ | split multiplicative | $38$ | \( 784 = 2^{4} \cdot 7^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 29008.b consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 7252.c1, its twist by $-4$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.1036.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.277983664.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | add | ord | ord | add | ord | ord | ord | ord | ord | ord | ord | split | ord | ord | ord |
| $\lambda$-invariant(s) | - | 2 | 2 | - | 2 | 2 | 2 | 6 | 2 | 2 | 2 | 3 | 2 | 2 | 2 |
| $\mu$-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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.