Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-95488x-11282708\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-95488xz^2-11282708z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-123753123x-524549730978\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-173, 397)$ | $3.7247814275297593357898080567$ | $\infty$ |
| $(-164, 82)$ | $0$ | $2$ |
| $(356, -178)$ | $0$ | $2$ |
Integral points
\( \left(-173, 397\right) \), \( \left(-173, -224\right) \), \( \left(-164, 82\right) \), \( \left(356, -178\right) \), \( \left(1526, 57542\right) \), \( \left(1526, -59068\right) \)
Invariants
| Conductor: | $N$ | = | \( 5070 \) | = | $2 \cdot 3 \cdot 5 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $1116653783976900$ | = | $2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 13^{10} $ |
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| j-invariant: | $j$ | = | \( \frac{19948814692561}{231344100} \) | = | $2^{-2} \cdot 3^{-4} \cdot 5^{-2} \cdot 13^{-4} \cdot 37^{3} \cdot 733^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6999743600499711971604999167$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.41749968131920282913375619592$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9729321235713593$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.393666472328214$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.7247814275297593357898080567$ |
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| Real period: | $\Omega$ | ≈ | $0.27177918632520894255343395573$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.0246361312265764246929558550 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.024636131 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.271779 \cdot 3.724781 \cdot 32}{4^2} \\ & \approx 2.024636131\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 43008 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
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 |
|---|---|---|
| $2$ | 2Cs | 8.24.0.14 |
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 $192$, genus $1$, and generators
$\left(\begin{array}{rr} 319 & 2 \\ 294 & 1555 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 479 & 1556 \\ 0 & 1559 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 1176 \\ 1554 & 385 \end{array}\right),\left(\begin{array}{rr} 521 & 8 \\ 524 & 33 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 1556 & 1557 \end{array}\right),\left(\begin{array}{rr} 1553 & 8 \\ 1552 & 9 \end{array}\right),\left(\begin{array}{rr} 3 & 398 \\ 382 & 1539 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$4830658560$ 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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 169 = 13^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 1690 = 2 \cdot 5 \cdot 13^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 5070.a
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 390.f3, its twist by $13$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-13}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-10}, \sqrt{13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{10}, \sqrt{13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1169858560000.7 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.592240896.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.94758543360000.59 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.0.8979181539709000089600000000.10 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | nonsplit | nonsplit | nonsplit | ss | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 2 | 1 | 1 | 1,1 | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | 1 | 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.