Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2+15742x+82113\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z+15742xz^2+82113z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+1275075x+56035125\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1348, 49725)$ | $0.28740387541304107872748156154$ | $\infty$ |
| $(73, 1275)$ | $0.36369137441661304144958169477$ | $\infty$ |
Integral points
\((-2,\pm 225)\), \((4,\pm 381)\), \((22,\pm 663)\), \((48,\pm 975)\), \((73,\pm 1275)\), \((124,\pm 1989)\), \((178,\pm 2925)\), \((328,\pm 6375)\), \((498,\pm 11475)\), \((529,\pm 12519)\), \((673,\pm 17775)\), \((1228,\pm 43275)\), \((1348,\pm 49725)\), \((3337,\pm 192933)\), \((3898,\pm 243525)\), \((7978,\pm 712725)\), \((10198,\pm 1029975)\), \((598048,\pm 462492225)\)
Invariants
| Conductor: | $N$ | = | \( 66300 \) | = | $2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $-252202713750000$ | = | $-1 \cdot 2^{4} \cdot 3^{5} \cdot 5^{7} \cdot 13^{2} \cdot 17^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{1725582942464}{1008810855} \) | = | $2^{8} \cdot 3^{-5} \cdot 5^{-1} \cdot 13^{-2} \cdot 17^{-3} \cdot 1889^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4528522584574398275618755169$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.41708424205374120378908514314$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9220035480723572$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.6575394311055667$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.091279997242550911061755967712$ |
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| Real period: | $\Omega$ | ≈ | $0.33521638639387606521927463863$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 360 $ = $ 3\cdot5\cdot2^{2}\cdot2\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $11.015478297248719670730522047 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.015478297 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.335216 \cdot 0.091280 \cdot 360}{1^2} \\ & \approx 11.015478297\end{aligned}$$
Modular invariants
Modular form 66300.2.a.w
For more coefficients, see the Downloads section to the right.
| Modular degree: | 241920 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $13$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $17$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 307 & 2 \\ 307 & 3 \end{array}\right),\left(\begin{array}{rr} 509 & 2 \\ 508 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 509 & 0 \end{array}\right),\left(\begin{array}{rr} 241 & 2 \\ 241 & 3 \end{array}\right),\left(\begin{array}{rr} 341 & 2 \\ 341 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[510])$ is a degree-$5414584320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/510\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 1275 = 3 \cdot 5^{2} \cdot 17 \) |
| $3$ | split multiplicative | $4$ | \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \) |
| $5$ | additive | $18$ | \( 884 = 2^{2} \cdot 13 \cdot 17 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 5100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 66300.w consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 13260.h1, its twist by $5$.
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 |
|---|---|---|---|
| $3$ | 3.1.255.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.16581375.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.1264873866750000.14 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | add | split | add | ord | ord | nonsplit | split | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 5 | - | 2 | 2 | 2 | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.