Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2+33430x+1058217\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z+33430xz^2+1058217z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+534885x+68260790\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(465, 10583\right) \) | $1.1998216639318426345947499566$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([465:10583:1]\) | $1.1998216639318426345947499566$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1859, 86528\right) \) | $1.1998216639318426345947499566$ | $\infty$ |
Integral points
\( \left(63, 1815\right) \), \( \left(63, -1879\right) \), \( \left(465, 10583\right) \), \( \left(465, -11049\right) \)
\([63:1815:1]\), \([63:-1879:1]\), \([465:10583:1]\), \([465:-11049:1]\)
\((251,\pm 14776)\), \((1859,\pm 86528)\)
Invariants
| Conductor: | $N$ | = | \( 76050 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-2882554889011200$ | = | $-1 \cdot 2^{15} \cdot 3^{6} \cdot 5^{2} \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{46969655}{32768} \) | = | $2^{-15} \cdot 5 \cdot 211^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6549873712549965266681635777$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.44503310388217674948966265041$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0629647337743247$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.8139247305045147$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.1998216639318426345947499566$ |
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| Real period: | $\Omega$ | ≈ | $0.28601151684898555214507520195$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 30 $ = $ ( 3 \cdot 5 )\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $10.294884421482602712680302813 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.294884421 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.286012 \cdot 1.199822 \cdot 30}{1^2} \\ & \approx 10.294884421\end{aligned}$$
Modular invariants
Modular form 76050.2.a.fj
For more coefficients, see the Downloads section to the right.
| Modular degree: | 388800 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $15$ | $I_{15}$ | split multiplicative | -1 | 1 | 15 | 15 |
| $3$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $1$ | $II$ | additive | 1 | 2 | 2 | 0 |
| $13$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
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 |
|---|---|---|---|
| $2$ | 2G | 8.2.0.1 | $2$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
| $5$ | 5B.4.1 | 5.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 781 & 1170 \\ 1365 & 1171 \end{array}\right),\left(\begin{array}{rr} 391 & 1170 \\ 975 & 1171 \end{array}\right),\left(\begin{array}{rr} 1 & 1482 \\ 1170 & 781 \end{array}\right),\left(\begin{array}{rr} 1039 & 390 \\ 1040 & 1559 \end{array}\right),\left(\begin{array}{rr} 479 & 0 \\ 0 & 1559 \end{array}\right),\left(\begin{array}{rr} 1041 & 650 \\ 520 & 937 \end{array}\right),\left(\begin{array}{rr} 1 & 312 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 521 & 1040 \\ 520 & 1041 \end{array}\right),\left(\begin{array}{rr} 1366 & 585 \\ 1365 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 780 & 1 \end{array}\right),\left(\begin{array}{rr} 481 & 1080 \\ 480 & 481 \end{array}\right),\left(\begin{array}{rr} 1 & 156 \\ 780 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1080 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 390 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$2415329280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $4$ | \( 38025 = 3^{2} \cdot 5^{2} \cdot 13^{2} \) |
| $3$ | additive | $6$ | \( 4225 = 5^{2} \cdot 13^{2} \) |
| $5$ | additive | $10$ | \( 1521 = 3^{2} \cdot 13^{2} \) |
| $13$ | additive | $86$ | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 5 and 15.
Its isogeny class 76050ek
consists of 4 curves linked by isogenies of
degrees dividing 15.
Twists
The minimal quadratic twist of this elliptic curve is 50b2, its twist by $-39$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{65}) \) | \(\Z/3\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-39}) \) | \(\Z/5\Z\) | not in database |
| $3$ | 3.1.200.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{-39})\) | \(\Z/15\Z\) | not in database |
| $6$ | 6.0.320000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.15015121875.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.2.439400000.6 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.2372760000.6 | \(\Z/10\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $12$ | deg 12 | \(\Z/15\Z\) | not in database |
| $12$ | deg 12 | \(\Z/30\Z\) | not in database |
| $18$ | 18.6.3369401822845625767214625000000000000.2 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.887414471860576333752000000000000000.5 | \(\Z/6\Z\) | not in database |
| $20$ | 20.4.37906719768380750901997089385986328125.1 | \(\Z/5\Z\) | 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 | add | add | ord | ord | add | ord | ord | ord | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | - | - | 1 | 1 | - | 1 | 1 | 1 | 1,1 | 1 | 1 | 3 | 1 | 1 |
| $\mu$-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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.