Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3-492750x-132404594\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3-492750xz^2-132404594z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-7884000x-8473894000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-77512090/203401, 16209323787/91733851)$ | $16.100633901878249565972009872$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 80325 \) | = | $3^{3} \cdot 5^{2} \cdot 7 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $83639786283703125$ | = | $3^{3} \cdot 5^{6} \cdot 7^{9} \cdot 17^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{31363160518656000}{198257271191} \) | = | $2^{15} \cdot 3^{9} \cdot 5^{3} \cdot 7^{-9} \cdot 17^{-3} \cdot 73^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0843497676966429593180774663$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0049777393125653491688864905$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1009571968779495$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.5101481797657765$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $16.100633901878249565972009872$ |
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| Real period: | $\Omega$ | ≈ | $0.18026369919444223565911668676$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot2\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.8047196530560391316348660910 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.804719653 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.180264 \cdot 16.100634 \cdot 2}{1^2} \\ & \approx 5.804719653\end{aligned}$$
Modular invariants
Modular form 80325.2.a.bt
For more coefficients, see the Downloads section to the right.
| Modular degree: | 715392 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $1$ | $II$ | additive | 1 | 3 | 3 | 0 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $1$ | $I_{9}$ | nonsplit multiplicative | 1 | 1 | 9 | 9 |
| $17$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 |
|---|---|---|
| $3$ | 3Cs | 9.36.0.3 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10710 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 10693 & 18 \\ 10692 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1261 & 2160 \\ 7065 & 8731 \end{array}\right),\left(\begin{array}{rr} 4283 & 0 \\ 0 & 10709 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 82 \end{array}\right),\left(\begin{array}{rr} 6431 & 2160 \\ 1935 & 6871 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 6121 & 2160 \\ 7965 & 8731 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[10710])$ is a degree-$12280277237760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10710\Z)$.
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 | $2$ | \( 25 = 5^{2} \) |
| $5$ | additive | $14$ | \( 3213 = 3^{3} \cdot 7 \cdot 17 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 11475 = 3^{3} \cdot 5^{2} \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 4725 = 3^{3} \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 80325.bt
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 3213.k2, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/3\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.3.12852.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{5})\) | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.6.58967083728.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.6.20646738000.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.61940214000.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.2892475694607268723741600330078125.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.7915146783240188021484375.1 | \(\Z/9\Z\) | 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 | add | nonsplit | ss | ord | nonsplit | ord | ord | ord | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | 2,13 | - | - | 1 | 1,1 | 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 | 0,0 | 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.