Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-5750x+155144\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-5750xz^2+155144z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-7451379x+7260764238\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(66, 220\right) \) | $0.82434440630219087027683895041$ | $\infty$ |
| \( \left(53, -27\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([66:220:1]\) | $0.82434440630219087027683895041$ | $\infty$ |
| \([53:-27:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(2379, 54756\right) \) | $0.82434440630219087027683895041$ | $\infty$ |
| \( \left(1911, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(53, -27\right) \), \( \left(66, 220\right) \), \( \left(66, -287\right) \), \( \left(78, 388\right) \), \( \left(78, -467\right) \), \( \left(261, 3925\right) \), \( \left(261, -4187\right) \)
\([53:-27:1]\), \([66:220:1]\), \([66:-287:1]\), \([78:388:1]\), \([78:-467:1]\), \([261:3925:1]\), \([261:-4187:1]\)
\( \left(1911, 0\right) \), \((2379,\pm 54756)\), \((2811,\pm 92340)\), \((9399,\pm 876096)\)
Invariants
| Conductor: | $N$ | = | \( 17238 \) | = | $2 \cdot 3 \cdot 13^{2} \cdot 17$ |
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| Minimal Discriminant: | $\Delta$ | = | $1701508094208$ | = | $2^{8} \cdot 3^{4} \cdot 13^{6} \cdot 17 $ |
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| j-invariant: | $j$ | = | \( \frac{4354703137}{352512} \) | = | $2^{-8} \cdot 3^{-4} \cdot 17^{-1} \cdot 23^{3} \cdot 71^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0898321935323226581387225953$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.19264248519844570988802112548$ |
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| $abc$ quality: | $Q$ | ≈ | $1.051915192059638$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.852866571845819$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.82434440630219087027683895041$ |
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| Real period: | $\Omega$ | ≈ | $0.82077763689773747222876452213$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.4128276303566447014900559472 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.412827630 \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 0.820778 \cdot 0.824344 \cdot 32}{2^2} \\ & \approx 5.412827630\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 36864 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $13$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $17$ | $1$ | $I_{1}$ | split 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 | 16.48.0.101 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3536 = 2^{4} \cdot 13 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 5 & 4 \\ 3532 & 3533 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 832 \\ 884 & 885 \end{array}\right),\left(\begin{array}{rr} 3521 & 16 \\ 3520 & 17 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 3438 & 3523 \end{array}\right),\left(\begin{array}{rr} 2744 & 273 \\ 3263 & 2458 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 271 & 0 \\ 0 & 3535 \end{array}\right),\left(\begin{array}{rr} 2289 & 832 \\ 234 & 495 \end{array}\right)$.
The torsion field $K:=\Q(E[3536])$ is a degree-$262787825664$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3536\Z)$.
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 | $4$ | \( 2873 = 13^{2} \cdot 17 \) |
| $3$ | split multiplicative | $4$ | \( 5746 = 2 \cdot 13^{2} \cdot 17 \) |
| $13$ | additive | $86$ | \( 102 = 2 \cdot 3 \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 17238f
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 102b1, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{17}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/8\Z\) | not in database |
| $2$ | \(\Q(\sqrt{221}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{13}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.176484635701504.43 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.176484635701504.39 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.2738521903104.13 | \(\Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | split | ord | ss | ord | add | split | ord | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 3 | 4 | 1 | 1,1 | 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 | 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.