Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-75171x+7900353\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-75171xz^2+7900353z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-97421643x+370060202502\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(147, 158\right) \) | $0.73132565880358301981957143469$ | $\infty$ |
| \( \left(-317, 158\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([147:158:1]\) | $0.73132565880358301981957143469$ | $\infty$ |
| \([-317:158:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(5307, 50112\right) \) | $0.73132565880358301981957143469$ | $\infty$ |
| \( \left(-11397, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-317, 158\right) \), \( \left(147, 158\right) \), \( \left(147, -306\right) \), \( \left(167, 114\right) \), \( \left(167, -282\right) \), \( \left(169, 158\right) \), \( \left(169, -328\right) \), \( \left(48283, 10585238\right) \), \( \left(48283, -10633522\right) \)
\([-317:158:1]\), \([147:158:1]\), \([147:-306:1]\), \([167:114:1]\), \([167:-282:1]\), \([169:158:1]\), \([169:-328:1]\), \([48283:10585238:1]\), \([48283:-10633522:1]\)
\( \left(-11397, 0\right) \), \((5307,\pm 50112)\), \((6027,\pm 42768)\), \((6099,\pm 52488)\), \((1738203,\pm 2291626080)\)
Invariants
| Conductor: | $N$ | = | \( 25230 \) | = | $2 \cdot 3 \cdot 5 \cdot 29^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $7373547832320$ | = | $2^{10} \cdot 3^{10} \cdot 5 \cdot 29^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{1926109896270461}{302330880} \) | = | $2^{-10} \cdot 3^{-10} \cdot 5^{-1} \cdot 11^{3} \cdot 11311^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4799501459512142993342925741$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.63812618845459579253847456601$ |
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| $abc$ quality: | $Q$ | ≈ | $1.022694830990156$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.468933407146121$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.73132565880358301981957143469$ |
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| Real period: | $\Omega$ | ≈ | $0.71895315582330614105884188864$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 40 $ = $ ( 2 \cdot 5 )\cdot2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.2578889033139444341060160971 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.257888903 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.718953 \cdot 0.731326 \cdot 40}{2^2} \\ & \approx 5.257888903\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 112000 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $3$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $29$ | $2$ | $III$ | 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 | 5.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3480 = 2^{3} \cdot 3 \cdot 5 \cdot 29 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 712 & 5 \\ 3435 & 3466 \end{array}\right),\left(\begin{array}{rr} 871 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2321 & 20 \\ 2330 & 201 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2298 & 5 \\ 3055 & 82 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 1741 & 20 \\ 10 & 201 \end{array}\right),\left(\begin{array}{rr} 3461 & 20 \\ 3460 & 21 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 3240 & 3131 \end{array}\right)$.
The torsion field $K:=\Q(E[3480])$ is a degree-$83813990400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3480\Z)$.
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 | $4$ | \( 145 = 5 \cdot 29 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 8410 = 2 \cdot 5 \cdot 29^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 841 = 29^{2} \) |
| $29$ | additive | $226$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 25230.p
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{145}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-319 +8 \sqrt{174}})\) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{290 +18 \sqrt{145}})\) | \(\Z/2\Z \oplus \Z/10\Z\) | 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 |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $20$ | 20.0.125571223144625902073297766037285327911376953125.4 | \(\Z/10\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | nonsplit | nonsplit | ord | ss | ord | ord | ord | ord | add | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 5 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.