Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-47913x+4025817\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-47913xz^2+4025817z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-62095275x+188014803750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(102, 399\right) \) | $0.10029912094957997008677652049$ | $\infty$ |
| \( \left(122, -61\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([102:399:1]\) | $0.10029912094957997008677652049$ | $\infty$ |
| \([122:-61:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(3675, 97200\right) \) | $0.10029912094957997008677652049$ | $\infty$ |
| \( \left(4395, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-198, 2499\right) \), \( \left(-198, -2301\right) \), \( \left(-78, 2739\right) \), \( \left(-78, -2661\right) \), \( \left(72, 939\right) \), \( \left(72, -1011\right) \), \( \left(102, 399\right) \), \( \left(102, -501\right) \), \( \left(122, -61\right) \), \( \left(132, 9\right) \), \( \left(132, -141\right) \), \( \left(138, 147\right) \), \( \left(138, -285\right) \), \( \left(186, 1155\right) \), \( \left(186, -1341\right) \), \( \left(282, 3459\right) \), \( \left(282, -3741\right) \), \( \left(606, 13755\right) \), \( \left(606, -14361\right) \), \( \left(2082, 93459\right) \), \( \left(2082, -95541\right) \)
\([-198:2499:1]\), \([-198:-2301:1]\), \([-78:2739:1]\), \([-78:-2661:1]\), \([72:939:1]\), \([72:-1011:1]\), \([102:399:1]\), \([102:-501:1]\), \([122:-61:1]\), \([132:9:1]\), \([132:-141:1]\), \([138:147:1]\), \([138:-285:1]\), \([186:1155:1]\), \([186:-1341:1]\), \([282:3459:1]\), \([282:-3741:1]\), \([606:13755:1]\), \([606:-14361:1]\), \([2082:93459:1]\), \([2082:-95541:1]\)
\((-7125,\pm 518400)\), \((-2805,\pm 583200)\), \((2595,\pm 210600)\), \((3675,\pm 97200)\), \( \left(4395, 0\right) \), \((4755,\pm 16200)\), \((4971,\pm 46656)\), \((6699,\pm 269568)\), \((10155,\pm 777600)\), \((21819,\pm 3036528)\), \((74955,\pm 20412000)\)
Invariants
| Conductor: | $N$ | = | \( 6450 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 43$ |
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| Minimal Discriminant: | $\Delta$ | = | $24074496000000$ | = | $2^{14} \cdot 3^{7} \cdot 5^{6} \cdot 43 $ |
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| j-invariant: | $j$ | = | \( \frac{778510269523657}{1540767744} \) | = | $2^{-14} \cdot 3^{-7} \cdot 11^{3} \cdot 43^{-1} \cdot 8363^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4555259874371868795179664080$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.65080703122013669221758674139$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0047908452218182$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.00978742990314$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.10029912094957997008677652049$ |
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| Real period: | $\Omega$ | ≈ | $0.67447891811309883621037536265$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 392 $ = $ ( 2 \cdot 7 )\cdot7\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.6296649734052193738304115562 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.629664973 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.674479 \cdot 0.100299 \cdot 392}{2^2} \\ & \approx 6.629664973\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 25088 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $14$ | $I_{14}$ | split multiplicative | -1 | 1 | 14 | 14 |
| $3$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $5$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $43$ | $1$ | $I_{1}$ | 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 | $\ell$-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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 3225 = 3 \cdot 5^{2} \cdot 43 \) |
| $3$ | split multiplicative | $4$ | \( 2150 = 2 \cdot 5^{2} \cdot 43 \) |
| $5$ | additive | $14$ | \( 258 = 2 \cdot 3 \cdot 43 \) |
| $7$ | good | $2$ | \( 1075 = 5^{2} \cdot 43 \) |
| $43$ | nonsplit multiplicative | $44$ | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 6450.bh
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 258.a1, its twist by $5$.
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 |
| $4$ | \(\Q(\sqrt{55 -10 \sqrt{-2}})\) | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.708922575360000.86 | \(\Z/2\Z \oplus \Z/4\Z\) | 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 | split | split | add | ord | ss | ord | ord | ord | ord | ord | ord | ord | ord | nonsplit | ord |
| $\lambda$-invariant(s) | 2 | 6 | - | 5 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 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.