Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-10513917x+13118341787\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-10513917xz^2+13118341787z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-851627304x+9565826044608\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(825469/441, 53432/9261)$ | $5.9686776638313823081208637374$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 18176 \) | = | $2^{8} \cdot 71$ |
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| Discriminant: | $\Delta$ | = | $2326528$ | = | $2^{15} \cdot 71 $ |
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| j-invariant: | $j$ | = | \( \frac{3922540634246430781376}{71} \) | = | $2^{6} \cdot 11^{3} \cdot 71^{-1} \cdot 358429^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1825160948607008563752614315$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.3160821191607692196037212797$ |
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| $abc$ quality: | $Q$ | ≈ | $1.043177284076972$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.12959918311655$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.9686776638313823081208637374$ |
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| Real period: | $\Omega$ | ≈ | $0.60658572100627712575166452373$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.2410292883384416001303182089 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.241029288 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.606586 \cdot 5.968678 \cdot 2}{1^2} \\ & \approx 7.241029288\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 147200 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | -1 | 8 | 15 | 0 |
| $71$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 |
|---|---|---|
| $5$ | 5B | 25.30.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 28400 = 2^{4} \cdot 5^{2} \cdot 71 \), index $1200$, genus $37$, and generators
$\left(\begin{array}{rr} 38 & 41 \\ 25841 & 25639 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 50 & 1 \end{array}\right),\left(\begin{array}{rr} 28351 & 50 \\ 28350 & 51 \end{array}\right),\left(\begin{array}{rr} 21289 & 28278 \\ 26801 & 4211 \end{array}\right),\left(\begin{array}{rr} 14140 & 7759 \\ 12277 & 24013 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 25211 & 20 \\ 13165 & 1391 \end{array}\right),\left(\begin{array}{rr} 3599 & 50 \\ 26025 & 1199 \end{array}\right)$.
The torsion field $K:=\Q(E[28400])$ is a degree-$153899827200000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/28400\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 71 \) |
| $71$ | split multiplicative | $72$ | \( 256 = 2^{8} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5 and 25.
Its isogeny class 18176.p
consists of 3 curves linked by isogenies of
degrees dividing 25.
Twists
The minimal quadratic twist of this elliptic curve is 18176.e1, its twist by $8$.
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 |
|---|---|---|---|
| $3$ | 3.3.568.1 | \(\Z/2\Z\) | not in database |
| $4$ | 4.4.256000.2 | \(\Z/5\Z\) | not in database |
| $6$ | 6.6.183250432.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/10\Z\) | not in database |
| $20$ | 20.0.1432792923035634298423738662876026293452800000000000000000000.1 | \(\Z/5\Z\) | not in database |
| $20$ | 20.20.43725369965687081861076008998902169600000000000000000000000000000000000.2 | \(\Z/25\Z\) | 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 | 71 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | split |
| $\lambda$-invariant(s) | - | 5 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 5 | 1 | 1 | 1 | 1 | 2 |
| $\mu$-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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.