Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3-x^2+3129x+10452\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3-x^2z+3129xz^2+10452z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+4054752x+536318928\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(108, 1263\right) \) | $0.70926642056852413254921420174$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([108:1263:1]\) | $0.70926642056852413254921420174$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(3876, 272916\right) \) | $0.70926642056852413254921420174$ | $\infty$ |
Integral points
\( \left(10, 206\right) \), \( \left(10, -207\right) \), \( \left(108, 1263\right) \), \( \left(108, -1264\right) \)
\([10:206:1]\), \([10:-207:1]\), \([108:1263:1]\), \([108:-1264:1]\)
\((348,\pm 44604)\), \((3876,\pm 272916)\)
Invariants
| Conductor: | $N$ | = | \( 12635 \) | = | $5 \cdot 7 \cdot 19^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-2017092147875$ | = | $-1 \cdot 5^{3} \cdot 7^{3} \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{71991296}{42875} \) | = | $2^{15} \cdot 5^{-3} \cdot 7^{-3} \cdot 13^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0503754688801701186145589991$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.42184402070305011138995471684$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0649344898971125$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.786301765833351$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.70926642056852413254921420174$ |
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| Real period: | $\Omega$ | ≈ | $0.50587185081895553881826153176$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 1\cdot3\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.1527875013804101099482694730 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.152787501 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.505872 \cdot 0.709266 \cdot 6}{1^2} \\ & \approx 2.152787501\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 14256 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $5$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $19$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 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 |
|---|---|---|---|
| $3$ | 3Cs | 3.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 11970 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 19 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 11953 & 18 \\ 11952 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 82 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1901 & 8208 \\ 9918 & 2395 \end{array}\right),\left(\begin{array}{rr} 9577 & 8208 \\ 513 & 2053 \end{array}\right),\left(\begin{array}{rr} 3779 & 0 \\ 0 & 11969 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7753 & 8208 \\ 3249 & 1141 \end{array}\right)$.
The torsion field $K:=\Q(E[11970])$ is a degree-$19300803379200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/11970\Z)$.
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 |
|---|---|---|---|
| $3$ | good | $2$ | \( 361 = 19^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 2527 = 7 \cdot 19^{2} \) |
| $7$ | split multiplicative | $8$ | \( 1805 = 5 \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 35 = 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 12635a
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 35a1, its twist by $-19$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{57}) \) | \(\Z/3\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-19}) \) | \(\Z/3\Z\) | 2.0.19.1-1225.5-a3 |
| $3$ | 3.1.140.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-19})\) | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.686000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.3629782800.2 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.134436400.2 | \(\Z/6\Z\) | 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 |
| $18$ | 18.6.8315202273275633574278288910905633056640625.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.1730379947738505061232732377131.1 | \(\Z/9\Z\) | 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 | ord | nonsplit | split | ord | ord | ord | add | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2,7 | 1 | 1 | 2 | 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 |
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.