Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-5582262x+5077966149\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-5582262xz^2+5077966149z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-89316187x+324900517366\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{7}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1377, 171\right) \) | $0.98205730514920074385434858962$ | $\infty$ |
| \( \left(257, 60371\right) \) | $0$ | $7$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1377:171:1]\) | $0.98205730514920074385434858962$ | $\infty$ |
| \([257:60371:1]\) | $0$ | $7$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(5507, 6880\right) \) | $0.98205730514920074385434858962$ | $\infty$ |
| \( \left(1027, 484000\right) \) | $0$ | $7$ |
Integral points
\( \left(-2493, 60371\right) \), \( \left(-2493, -57879\right) \), \( \left(257, 60371\right) \), \( \left(257, -60629\right) \), \( \left(907, 27121\right) \), \( \left(907, -28029\right) \), \( \left(1357, -129\right) \), \( \left(1357, -1229\right) \), \( \left(1377, 171\right) \), \( \left(1377, -1549\right) \), \( \left(1467, 5921\right) \), \( \left(1467, -7389\right) \), \( \left(1643, 17603\right) \), \( \left(1643, -19247\right) \), \( \left(2237, 60371\right) \), \( \left(2237, -62609\right) \), \( \left(4129, 226867\right) \), \( \left(4129, -230997\right) \), \( \left(17197, 2226271\right) \), \( \left(17197, -2243469\right) \)
\([-2493:60371:1]\), \([-2493:-57879:1]\), \([257:60371:1]\), \([257:-60629:1]\), \([907:27121:1]\), \([907:-28029:1]\), \([1357:-129:1]\), \([1357:-1229:1]\), \([1377:171:1]\), \([1377:-1549:1]\), \([1467:5921:1]\), \([1467:-7389:1]\), \([1643:17603:1]\), \([1643:-19247:1]\), \([2237:60371:1]\), \([2237:-62609:1]\), \([4129:226867:1]\), \([4129:-230997:1]\), \([17197:2226271:1]\), \([17197:-2243469:1]\)
\((-9973,\pm 473000)\), \((1027,\pm 484000)\), \((3627,\pm 220600)\), \((5427,\pm 4400)\), \((5507,\pm 6880)\), \((5867,\pm 53240)\), \((6571,\pm 147400)\), \((8947,\pm 491920)\), \((16515,\pm 1831456)\), \((68787,\pm 17878960)\)
Invariants
| Conductor: | $N$ | = | \( 4730 \) | = | $2 \cdot 5 \cdot 11 \cdot 43$ |
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| Minimal Discriminant: | $\Delta$ | = | $-360317791790000000$ | = | $-1 \cdot 2^{7} \cdot 5^{7} \cdot 11^{7} \cdot 43^{2} $ |
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| j-invariant: | $j$ | = | \( -\frac{19237750463016353596082481}{360317791790000000} \) | = | $-1 \cdot 2^{-7} \cdot 3^{3} \cdot 5^{-7} \cdot 11^{-7} \cdot 37^{3} \cdot 43^{-2} \cdot 2413951^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4932317077383535936535959131$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.4932317077383535936535959131$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0231268804034592$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.88030577877314$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.98205730514920074385434858962$ |
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| Real period: | $\Omega$ | ≈ | $0.27814414800151682937355721907$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 686 $ = $ 7\cdot7\cdot7\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $7$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.8241488940114609400622108843 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.824148894 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.278144 \cdot 0.982057 \cdot 686}{7^2} \\ & \approx 3.824148894\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 219520 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $5$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $11$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $43$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 |
|---|---|---|---|
| $7$ | 7B.1.1 | 7.48.0.1 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3080 = 2^{3} \cdot 5 \cdot 7 \cdot 11 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3067 & 14 \\ 3066 & 15 \end{array}\right),\left(\begin{array}{rr} 1856 & 7 \\ 1841 & 3074 \end{array}\right),\left(\begin{array}{rr} 771 & 1554 \\ 0 & 2971 \end{array}\right),\left(\begin{array}{rr} 2311 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 1688 & 7 \\ 833 & 3074 \end{array}\right),\left(\begin{array}{rr} 8 & 7 \\ 1533 & 3074 \end{array}\right)$.
The torsion field $K:=\Q(E[3080])$ is a degree-$204374016000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3080\Z)$.
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 | $4$ | \( 55 = 5 \cdot 11 \) |
| $5$ | split multiplicative | $6$ | \( 946 = 2 \cdot 11 \cdot 43 \) |
| $7$ | good | $2$ | \( 43 \) |
| $11$ | split multiplicative | $12$ | \( 430 = 2 \cdot 5 \cdot 43 \) |
| $43$ | split multiplicative | $44$ | \( 110 = 2 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 4730k
consists of 2 curves linked by isogenies of
degree 7.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.440.1 | \(\Z/14\Z\) | not in database |
| $6$ | 6.0.85184000.1 | \(\Z/2\Z \oplus \Z/14\Z\) | not in database |
| $8$ | deg 8 | \(\Z/21\Z\) | not in database |
| $12$ | deg 12 | \(\Z/28\Z\) | 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 | split | ss | split | ord | split | ss | ord | ord | ord | ord | ord | ord | ss | split | ord |
| $\lambda$-invariant(s) | 3 | 1,1 | 4 | 7 | 2 | 1,1 | 1 | 1 | 3 | 1 | 1 | 1 | 1,1 | 2 | 1 |
| $\mu$-invariant(s) | 0 | 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.