Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-5510855x-17315148075\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-5510855xz^2-17315148075z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-7142068107x-807834122382906\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{13135}{4}, -\frac{13135}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([26270:-13135:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(118218, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 10230 \) | = | $2 \cdot 3 \cdot 5 \cdot 11 \cdot 31$ |
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| Minimal Discriminant: | $\Delta$ | = | $-118801759721890483665900$ | = | $-1 \cdot 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11^{5} \cdot 31^{10} $ |
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| j-invariant: | $j$ | = | \( -\frac{18508902577171306222471921}{118801759721890483665900} \) | = | $-1 \cdot 2^{-2} \cdot 3^{-2} \cdot 5^{-2} \cdot 11^{-5} \cdot 31^{-10} \cdot 149^{3} \cdot 1775309^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1111473403413294567959790425$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.1111473403413294567959790425$ |
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| $abc$ quality: | $Q$ | ≈ | $1.028999293653901$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.571240425985704$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
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: | $\Omega$ | ≈ | $0.043919344540703761019506075844$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 400 $ = $ 2\cdot2\cdot2\cdot5\cdot( 2 \cdot 5 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $4.3919344540703761019506075844 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 4.391934454 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.043919 \cdot 1.000000 \cdot 400}{2^2} \\ & \approx 4.391934454\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1760000 |
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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 semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $11$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $31$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
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 | 2.3.0.1 | $3$ |
| $5$ | 5B.1.2 | 5.24.0.3 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 20460 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \cdot 31 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 11 & 16 \\ 20220 & 20111 \end{array}\right),\left(\begin{array}{rr} 12277 & 20 \\ 10 & 201 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9241 & 20 \\ 10570 & 201 \end{array}\right),\left(\begin{array}{rr} 20441 & 20 \\ 20440 & 21 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 6821 & 20 \\ 6830 & 201 \end{array}\right),\left(\begin{array}{rr} 10231 & 20 \\ 10 & 201 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 9316 & 5 \\ 18555 & 20446 \end{array}\right)$.
The torsion field $K:=\Q(E[20460])$ is a degree-$90508492800000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/20460\Z)$.
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 | $4$ | \( 11 \) |
| $3$ | split multiplicative | $4$ | \( 3410 = 2 \cdot 5 \cdot 11 \cdot 31 \) |
| $5$ | split multiplicative | $6$ | \( 6 = 2 \cdot 3 \) |
| $11$ | split multiplicative | $12$ | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
| $31$ | split multiplicative | $32$ | \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 10230.bf
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-11}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-74 +6 \sqrt{465}})\) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\zeta_{5})\) | \(\Z/10\Z\) | not in database |
| $5$ | 5.1.2531250000.1 | \(\Z/10\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.228765625.1 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $10$ | 10.0.1031890245117187500000000.3 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/20\Z\) | not in database |
| $20$ | 20.0.5131569027900695800781250000000000000000.1 | \(\Z/5\Z \oplus \Z/10\Z\) | not in database |
| $20$ | 20.2.4994166215876227776290092298730468750000000000000000000000000000.1 | \(\Z/20\Z\) | 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 | 11 | 31 |
|---|---|---|---|---|---|
| Reduction type | split | split | split | split | split |
| $\lambda$-invariant(s) | 1 | 1 | 3 | 1 | 1 |
| $\mu$-invariant(s) | 1 | 0 | 1 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.