Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-78624005x-268317472003\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-78624005xz^2-268317472003z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1257984075x-17173576192250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-5121, 2560)$ | $0$ | $2$ |
| $(10239, -5120)$ | $0$ | $2$ |
Integral points
\( \left(-5121, 2560\right) \), \( \left(10239, -5120\right) \)
Invariants
| Conductor: | $N$ | = | \( 40950 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $12518010086400000000$ | = | $2^{16} \cdot 3^{10} \cdot 5^{8} \cdot 7^{2} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{4718909406724749250561}{1098974822400} \) | = | $2^{-16} \cdot 3^{-4} \cdot 5^{-2} \cdot 7^{-2} \cdot 13^{-2} \cdot 433^{3} \cdot 38737^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0443026219416232825178018607$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.6902775213905182495197995756$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0081936394487907$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.229143013186678$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.050700328188246552467256082691$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1024 $ = $ 2^{4}\cdot2^{2}\cdot2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.2448210040477793579043892922 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 3.244821004 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.050700 \cdot 1.000000 \cdot 1024}{4^2} \\ & \approx 3.244821004\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3145728 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $16$ | $I_{16}$ | split multiplicative | -1 | 1 | 16 | 16 |
| $3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $5$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $2$ | $I_{2}$ | nonsplit 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 |
|---|---|---|
| $2$ | 2Cs | 16.48.0.3 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 21840 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 17365 & 14568 \\ 19518 & 21709 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21825 & 16 \\ 21824 & 17 \end{array}\right),\left(\begin{array}{rr} 17459 & 7272 \\ 7350 & 14603 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 18721 & 7296 \\ 3126 & 97 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 4 & 65 \end{array}\right),\left(\begin{array}{rr} 7285 & 7296 \\ 7284 & 5473 \end{array}\right),\left(\begin{array}{rr} 14575 & 7296 \\ 14784 & 21169 \end{array}\right),\left(\begin{array}{rr} 7279 & 0 \\ 0 & 21839 \end{array}\right)$.
The torsion field $K:=\Q(E[21840])$ is a degree-$38954430627840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/21840\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 225 = 3^{2} \cdot 5^{2} \) |
| $3$ | additive | $8$ | \( 4550 = 2 \cdot 5^{2} \cdot 7 \cdot 13 \) |
| $5$ | additive | $18$ | \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 40950.ef
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 2730.w5, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{15}, \sqrt{-273})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{15}, \sqrt{273})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{-39})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.888731494560000.169 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.31116960000.7 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.6108930140625.2 | \(\Z/2\Z \oplus \Z/16\Z\) | 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 |
| $16$ | 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 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 13 |
|---|---|---|---|---|---|
| Reduction type | split | add | add | split | nonsplit |
| $\lambda$-invariant(s) | 5 | - | - | 3 | 0 |
| $\mu$-invariant(s) | 0 | - | - | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.