Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-10082738x+12325121014\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-10082738xz^2+12325121014z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-13067227827x+575080047724302\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1474, 25074\right) \) | $1.8858829817601642358250150967$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1474:25074:1]\) | $1.8858829817601642358250150967$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(53067, 5575284\right) \) | $1.8858829817601642358250150967$ | $\infty$ |
Integral points
\( \left(1474, 25074\right) \), \( \left(1474, -26549\right) \)
\([1474:25074:1]\), \([1474:-26549:1]\)
\((53067,\pm 5575284)\)
Invariants
| Conductor: | $N$ | = | \( 103246 \) | = | $2 \cdot 11 \cdot 13 \cdot 19^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-31726473771853080506$ | = | $-1 \cdot 2 \cdot 11^{10} \cdot 13 \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{2409558590804994721}{674373039626} \) | = | $-1 \cdot 2^{-1} \cdot 11^{-10} \cdot 13^{-1} \cdot 29^{3} \cdot 46229^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7234749030731995039877140910$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.2512554134899792739832003751$ |
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| $abc$ quality: | $Q$ | ≈ | $1.104756801261857$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.196515320433505$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.8858829817601642358250150967$ |
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| Real period: | $\Omega$ | ≈ | $0.20346143330252685362421864301$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 20 $ = $ 1\cdot( 2 \cdot 5 )\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.6740890901953224502333942318 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.674089090 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.203461 \cdot 1.885883 \cdot 20}{1^2} \\ & \approx 7.674089090\end{aligned}$$
Modular invariants
Modular form 103246.2.a.g
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4320000 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $11$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $13$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $19$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|---|
| $5$ | 5B.4.2 | 5.12.0.2 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9880 = 2^{3} \cdot 5 \cdot 13 \cdot 19 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 4941 & 2090 \\ 5985 & 571 \end{array}\right),\left(\begin{array}{rr} 8361 & 2090 \\ 3325 & 571 \end{array}\right),\left(\begin{array}{rr} 2471 & 2090 \\ 3515 & 571 \end{array}\right),\left(\begin{array}{rr} 4159 & 0 \\ 0 & 9879 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 2737 & 7030 \\ 1520 & 9121 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 9825 & 9761 \end{array}\right),\left(\begin{array}{rr} 9871 & 10 \\ 9870 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[9880])$ is a degree-$49562556825600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9880\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 4693 = 13 \cdot 19^{2} \) |
| $5$ | good | $2$ | \( 9386 = 2 \cdot 13 \cdot 19^{2} \) |
| $11$ | split multiplicative | $12$ | \( 9386 = 2 \cdot 13 \cdot 19^{2} \) |
| $13$ | nonsplit multiplicative | $14$ | \( 7942 = 2 \cdot 11 \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 286 = 2 \cdot 11 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 103246g
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 286d2, 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 |
|---|---|---|---|
| $3$ | 3.1.104.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{190 -38 \sqrt{5}})\) | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.1124864.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $10$ | 10.0.5049575056343447500000000.1 | \(\Z/5\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/10\Z\) | 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 | nonsplit | ord | ord | ord | split | nonsplit | ord | add | ord | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | 1 | 3 | 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.