Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-4319x-100435\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-4319xz^2-100435z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-349866x-72167544\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-31, 68)$ | $3.4580401217040249036284953379$ | $\infty$ |
Integral points
\((-31,\pm 68)\)
Invariants
| Conductor: | $N$ | = | \( 18176 \) | = | $2^{8} \cdot 71$ |
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| Discriminant: | $\Delta$ | = | $923765427712$ | = | $2^{9} \cdot 71^{5} $ |
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| j-invariant: | $j$ | = | \( \frac{17406197775296}{1804229351} \) | = | $2^{6} \cdot 11^{3} \cdot 19^{3} \cdot 31^{3} \cdot 71^{-5}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0312235483636780143662657042$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.51136316294371903230334161311$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9556798643741012$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.744566307786992$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.4580401217040249036284953379$ |
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| Real period: | $\Omega$ | ≈ | $0.59284807951587900061533379731$ |
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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)$ | ≈ | $4.1001848900821753055134264267 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.100184890 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.592848 \cdot 3.458040 \cdot 2}{1^2} \\ & \approx 4.100184890\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 14720 |
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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 | 9 | 0 |
| $71$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
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 |
|---|---|---|
| $5$ | 5Cs | 5.30.0.1 |
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
$\left(\begin{array}{rr} 28351 & 50 \\ 28350 & 51 \end{array}\right),\left(\begin{array}{rr} 16 & 35 \\ 1765 & 3861 \end{array}\right),\left(\begin{array}{rr} 16201 & 50 \\ 9504 & 11977 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 10 & 501 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7143 & 50 \\ 25821 & 5257 \end{array}\right),\left(\begin{array}{rr} 16831 & 50 \\ 21535 & 23121 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 50 & 1 \end{array}\right),\left(\begin{array}{rr} 24801 & 28350 \\ 2375 & 27201 \end{array}\right)$.
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 | $4$ | \( 71 \) |
| $5$ | good | $2$ | \( 256 = 2^{8} \) |
| $71$ | nonsplit multiplicative | $72$ | \( 256 = 2^{8} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 18176b
consists of 3 curves linked by isogenies of
degrees dividing 25.
Twists
The minimal quadratic twist of this elliptic curve is 18176r2, its twist by $-4$.
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 |
| $4$ | \(\Q(\zeta_{16})^+\) | \(\Z/5\Z\) | not in database |
| $4$ | 4.0.256000.4 | \(\Z/5\Z\) | 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 |
| $12$ | deg 12 | \(\Z/10\Z\) | not in database |
| $16$ | 16.0.4294967296000000000000.1 | \(\Z/5\Z \oplus \Z/5\Z\) | 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 | nonsplit |
| $\lambda$-invariant(s) | - | 3 | 1 | 3 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 1 |
| $\mu$-invariant(s) | - | 0 | 1 | 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.