Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-828x+9072\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-828xz^2+9072z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1073115x+426482550\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{10}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(12, 24\right) \) | $0$ | $10$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([12:24:1]\) | $0$ | $10$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(435, 6480\right) \) | $0$ | $10$ |
Integral points
\( \left(-18, 144\right) \), \( \left(-18, -126\right) \), \( \left(12, 24\right) \), \( \left(12, -36\right) \), \( \left(18, 0\right) \), \( \left(18, -18\right) \), \( \left(36, 144\right) \), \( \left(36, -180\right) \)
\([-18:144:1]\), \([-18:-126:1]\), \([12:24:1]\), \([12:-36:1]\), \([18:0:1]\), \([18:-18:1]\), \([36:144:1]\), \([36:-180:1]\)
\((-645,\pm 29160)\), \((435,\pm 6480)\), \((651,\pm 1944)\), \((1299,\pm 34992)\)
Invariants
| Conductor: | $N$ | = | \( 150 \) | = | $2 \cdot 3 \cdot 5^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $236196000$ | = | $2^{5} \cdot 3^{10} \cdot 5^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{502270291349}{1889568} \) | = | $2^{-5} \cdot 3^{-10} \cdot 7949^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.46602136388289199159133673149$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.063661885774366897941146898184$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0757503122978644$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.340656645965683$ | |||
| 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: | $\Omega$ | ≈ | $1.7692592482524138311369605540$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 100 $ = $ 5\cdot( 2 \cdot 5 )\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $10$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.7692592482524138311369605540 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.769259248 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.769259 \cdot 1.000000 \cdot 100}{10^2} \\ & \approx 1.769259248\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 80 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $3$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $5$ | $2$ | $III$ | additive | -1 | 2 | 3 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
| $5$ | 5B.1.1 | 5.24.0.1 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120 = 2^{3} \cdot 3 \cdot 5 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 11 & 16 \\ 0 & 11 \end{array}\right),\left(\begin{array}{rr} 101 & 20 \\ 100 & 21 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 31 & 20 \\ 70 & 81 \end{array}\right),\left(\begin{array}{rr} 41 & 20 \\ 50 & 81 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 66 & 5 \\ 55 & 82 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 16 & 5 \\ 15 & 106 \end{array}\right)$.
The torsion field $K:=\Q(E[120])$ is a degree-$122880$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.
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 | $4$ | \( 5 \) |
| $3$ | split multiplicative | $4$ | \( 50 = 2 \cdot 5^{2} \) |
| $5$ | additive | $10$ | \( 1 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 150.c
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 $E(\Q)_{\rm tors}$ $\cong \Z/{10}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{10}) \) | \(\Z/2\Z \oplus \Z/10\Z\) | 2.2.40.1-450.1-bh5 |
| $4$ | \(\Q(\sqrt{10 +2 \sqrt{-15}})\) | \(\Z/20\Z\) | not in database |
| $8$ | 8.4.589824000000.9 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $8$ | 8.0.82944000000.10 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $8$ | 8.2.44286750000.2 | \(\Z/30\Z\) | not in database |
| $16$ | deg 16 | \(\Z/40\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/30\Z\) | not in database |
| $20$ | 20.0.4656612873077392578125.1 | \(\Z/5\Z \oplus \Z/10\Z\) | 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 |
|---|---|---|---|
| Reduction type | split | split | add |
| $\lambda$-invariant(s) | 1 | 1 | - |
| $\mu$-invariant(s) | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ 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 $0$.