Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-942x-6409\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-942xz^2-6409z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-15075x-425250\)
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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 |
|---|---|---|
| \( \left(-10, 49\right) \) | $2.1574950188562100247262909349$ | $\infty$ |
| \( \left(-26, 13\right) \) | $0$ | $2$ |
| \( \left(34, -17\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-10:49:1]\) | $2.1574950188562100247262909349$ | $\infty$ |
| \([-26:13:1]\) | $0$ | $2$ |
| \([34:-17:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-41, 352\right) \) | $2.1574950188562100247262909349$ | $\infty$ |
| \( \left(-105, 0\right) \) | $0$ | $2$ |
| \( \left(135, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-26, 13\right) \), \( \left(-10, 49\right) \), \( \left(-10, -39\right) \), \( \left(34, -17\right) \), \( \left(274, 4363\right) \), \( \left(274, -4637\right) \)
\([-26:13:1]\), \([-10:49:1]\), \([-10:-39:1]\), \([34:-17:1]\), \([274:4363:1]\), \([274:-4637:1]\)
\( \left(-105, 0\right) \), \((-41,\pm 352)\), \( \left(135, 0\right) \), \((1095,\pm 36000)\)
Invariants
| Conductor: | $N$ | = | \( 2475 \) | = | $3^{2} \cdot 5^{2} \cdot 11$ |
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| Minimal Discriminant: | $\Delta$ | = | $34456640625$ | = | $3^{6} \cdot 5^{8} \cdot 11^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{8120601}{3025} \) | = | $3^{3} \cdot 5^{-2} \cdot 11^{-2} \cdot 67^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.72178603029637347762639881471$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.63223907025473155537160347036$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0555982735754514$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.115463748420747$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.1574950188562100247262909349$ |
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| Real period: | $\Omega$ | ≈ | $0.88886178683078034013809041399$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.8354297550780779307618261547 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.835429755 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.888862 \cdot 2.157495 \cdot 32}{4^2} \\ & \approx 3.835429755\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1536 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $11$ | $2$ | $I_{2}$ | 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 |
|---|---|---|---|
| $2$ | 2Cs | 2.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 660 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 439 & 0 \\ 0 & 659 \end{array}\right),\left(\begin{array}{rr} 553 & 444 \\ 228 & 451 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 569 & 438 \\ 354 & 221 \end{array}\right),\left(\begin{array}{rr} 541 & 222 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 657 & 4 \\ 656 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[660])$ is a degree-$608256000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/660\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 225 = 3^{2} \cdot 5^{2} \) |
| $3$ | additive | $6$ | \( 275 = 5^{2} \cdot 11 \) |
| $5$ | additive | $18$ | \( 99 = 3^{2} \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 225 = 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 2475.i
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 55.a3, its twist by $-15$.
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 |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{3}, \sqrt{-55})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{11}, \sqrt{15})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{5})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.500310421875.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | \(\Q(i, \sqrt{3}, \sqrt{5}, \sqrt{11})\) | \(\Z/4\Z \oplus \Z/4\Z\) | 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 |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | ord | add | add | ss | split | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | - | - | 1,1 | 2 | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 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.