Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-65387463x+219523281417\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-65387463xz^2+219523281417z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-84742152075x+10242332444247750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-3018, 625509)$ | $0.32041674133029897047491204025$ | $\infty$ |
| $(-9418, 4709)$ | $0$ | $2$ |
Integral points
\( \left(-9418, 4709\right) \), \( \left(-3018, 625509\right) \), \( \left(-3018, -622491\right) \), \( \left(1818, 325677\right) \), \( \left(1818, -327495\right) \), \( \left(2832, 237459\right) \), \( \left(2832, -240291\right) \), \( \left(4982, 129509\right) \), \( \left(4982, -134491\right) \), \( \left(6966, 316005\right) \), \( \left(6966, -322971\right) \), \( \left(21942, 3046629\right) \), \( \left(21942, -3068571\right) \)
Invariants
| Conductor: | $N$ | = | \( 95550 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-2927129696560742400000000$ | = | $-1 \cdot 2^{30} \cdot 3^{3} \cdot 5^{8} \cdot 7^{6} \cdot 13^{3} $ |
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| j-invariant: | $j$ | = | \( -\frac{16818951115904497561}{1592332281446400} \) | = | $-1 \cdot 2^{-30} \cdot 3^{-3} \cdot 5^{-2} \cdot 13^{-3} \cdot 17^{3} \cdot 29^{3} \cdot 5197^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.4364864940456754898576588954$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.6588124633009686500046028571$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0346538273212997$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.7338729841801825$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.32041674133029897047491204025$ |
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| Real period: | $\Omega$ | ≈ | $0.078427978452949760148061560343$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2160 $ = $ ( 2 \cdot 3 \cdot 5 )\cdot3\cdot2^{2}\cdot2\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $13.570004133909214842318184691 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 13.570004134 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.078428 \cdot 0.320417 \cdot 2160}{2^2} \\ & \approx 13.570004134\end{aligned}$$
Modular invariants
Modular form 95550.2.a.ke
For more coefficients, see the Downloads section to the right.
| Modular degree: | 18662400 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $30$ | $I_{30}$ | split multiplicative | -1 | 1 | 30 | 30 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $5$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5461 & 7812 \\ 9366 & 3193 \end{array}\right),\left(\begin{array}{rr} 3119 & 0 \\ 0 & 10919 \end{array}\right),\left(\begin{array}{rr} 6315 & 9688 \\ 4718 & 1877 \end{array}\right),\left(\begin{array}{rr} 1310 & 4683 \\ 5173 & 7792 \end{array}\right),\left(\begin{array}{rr} 10909 & 12 \\ 10908 & 13 \end{array}\right),\left(\begin{array}{rr} 1450 & 4683 \\ 8253 & 7792 \end{array}\right),\left(\begin{array}{rr} 2183 & 3108 \\ 3738 & 7727 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 10870 & 10911 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$19477215313920$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
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 | $4$ | \( 47775 = 3 \cdot 5^{2} \cdot 7^{2} \cdot 13 \) |
| $3$ | split multiplicative | $4$ | \( 1225 = 5^{2} \cdot 7^{2} \) |
| $5$ | additive | $18$ | \( 1911 = 3 \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $26$ | \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 95550jo
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 390d3, its twist by $-35$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-39}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{105}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.3057600.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-39}, \sqrt{105})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.781396875.1 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.14219703912960000.223 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.84140259840000.23 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.35846476177713434462726119055942145000000000000.1 | \(\Z/18\Z\) | 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 | add | ss | split | ss | ord | ord | ss | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 6 | 6 | - | - | 3,1 | 2 | 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 | 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.