Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3-124509x+18375453\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3-124509xz^2+18375453z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1992144x+1176029008\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(10241/16, 907105/64)$ | $3.1988253742073634444065770244$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 53361 \) | = | $3^{2} \cdot 7^{2} \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $-22335167517477507$ | = | $-1 \cdot 3^{7} \cdot 7^{8} \cdot 11^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{28672}{3} \) | = | $-1 \cdot 2^{12} \cdot 3^{-1} \cdot 7$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8761213388418444682562856272$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.1694058745949378528758772759$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9123874511245065$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.315747207866878$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.1988253742073634444065770244$ |
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| Real period: | $\Omega$ | ≈ | $0.37148759022907294701806339679$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2\cdot3\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $14.259887157934871540190461571 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.259887158 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.371488 \cdot 3.198825 \cdot 12}{1^2} \\ & \approx 14.259887158\end{aligned}$$
Modular invariants
Modular form 53361.2.a.cd
For more coefficients, see the Downloads section to the right.
| Modular degree: | 470400 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $7$ | $3$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
| $11$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
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 |
|---|---|---|
| $13$ | 13B.4.1 | 13.28.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6006 = 2 \cdot 3 \cdot 7 \cdot 11 \cdot 13 \), index $336$, genus $9$, and generators
$\left(\begin{array}{rr} 14 & 23 \\ 871 & 1431 \end{array}\right),\left(\begin{array}{rr} 1 & 572 \\ 0 & 925 \end{array}\right),\left(\begin{array}{rr} 1637 & 0 \\ 0 & 6005 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 26 & 1 \end{array}\right),\left(\begin{array}{rr} 5125 & 5434 \\ 4290 & 4663 \end{array}\right),\left(\begin{array}{rr} 4718 & 2717 \\ 5291 & 3288 \end{array}\right),\left(\begin{array}{rr} 1 & 26 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5981 & 26 \\ 5980 & 27 \end{array}\right)$.
The torsion field $K:=\Q(E[6006])$ is a degree-$597793996800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6006\Z)$.
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 |
|---|---|---|---|
| $3$ | additive | $8$ | \( 5929 = 7^{2} \cdot 11^{2} \) |
| $7$ | additive | $26$ | \( 1089 = 3^{2} \cdot 11^{2} \) |
| $11$ | additive | $62$ | \( 441 = 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
13.
Its isogeny class 53361u
consists of 2 curves linked by isogenies of
degree 13.
Twists
The minimal quadratic twist of this elliptic curve is 147c1, its twist by $-231$.
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 |
|---|---|---|---|
| $3$ | 3.1.588.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1037232.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.6.86284737.1 | \(\Z/13\Z\) | not in database |
| $8$ | 8.2.6227255754027.2 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $18$ | 18.6.2631248629891740460251353088.1 | \(\Z/26\Z\) | 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 | ss | add | ord | add | add | ord | ss | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | ? | - | 3 | - | - | 3 | 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 |
An entry ? indicates that the invariants have not yet been computed.
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.