Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-3743701059x-95785856035774\)
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(homogenize, simplify) |
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\(y^2z=x^3-3743701059xz^2-95785856035774z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-3743701059x-95785856035774\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 100368 \) | = | $2^{4} \cdot 3^{2} \cdot 17 \cdot 41$ |
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| Discriminant: | $\Delta$ | = | $-605548443031981490018288664576$ | = | $-1 \cdot 2^{18} \cdot 3^{9} \cdot 17^{15} \cdot 41 $ |
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| j-invariant: | $j$ | = | \( -\frac{1943299427371886688757286977}{202796948353367429302464} \) | = | $-1 \cdot 2^{-6} \cdot 3^{-3} \cdot 7^{3} \cdot 17^{-15} \cdot 41^{-1} \cdot 79^{3} \cdot 2256601^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.4537761039298401323098454479$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.2113227790358399771949907080$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0364691817074154$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.764967660686344$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.0095937550056734701972234378294$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 240 $ = $ 2^{2}\cdot2^{2}\cdot( 3 \cdot 5 )\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.3025012013616328473336250790 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.302501201 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.009594 \cdot 1.000000 \cdot 240}{1^2} \\ & \approx 2.302501201\end{aligned}$$
Modular invariants
Modular form 100368.2.a.g
For more coefficients, see the Downloads section to the right.
| Modular degree: | 153031680 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{10}^{*}$ | additive | -1 | 4 | 18 | 6 |
| $3$ | $4$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $17$ | $15$ | $I_{15}$ | split multiplicative | -1 | 1 | 15 | 15 |
| $41$ | $1$ | $I_{1}$ | nonsplit 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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8364 = 2^{2} \cdot 3 \cdot 17 \cdot 41 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 4187 & 8358 \\ 4188 & 8357 \end{array}\right),\left(\begin{array}{rr} 4181 & 0 \\ 0 & 8363 \end{array}\right),\left(\begin{array}{rr} 5201 & 8358 \\ 7239 & 8345 \end{array}\right),\left(\begin{array}{rr} 5165 & 8358 \\ 7131 & 8345 \end{array}\right),\left(\begin{array}{rr} 8359 & 6 \\ 8358 & 7 \end{array}\right),\left(\begin{array}{rr} 4876 & 7665 \\ 8363 & 6272 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[8364])$ is a degree-$62159427993600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8364\Z)$.
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 | $2$ | \( 6273 = 3^{2} \cdot 17 \cdot 41 \) |
| $3$ | additive | $6$ | \( 656 = 2^{4} \cdot 41 \) |
| $5$ | good | $2$ | \( 5904 = 2^{4} \cdot 3^{2} \cdot 41 \) |
| $17$ | split multiplicative | $18$ | \( 5904 = 2^{4} \cdot 3^{2} \cdot 41 \) |
| $41$ | nonsplit multiplicative | $42$ | \( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 100368bt
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 4182h2, its twist by $12$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.2091.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.9142439571.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.395516115648.4 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.279825984.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.3099919777798894608730979203085951336355987456.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.2510462817466332754609119718272340003029516288.2 | \(\Z/6\Z\) | 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 | add | add | ord | ord | ord | ord | split | ord | ss | ord | ord | ord | nonsplit | ord | ord |
| $\lambda$-invariant(s) | - | - | 4 | 6 | 0 | 0 | 1 | 0 | 0,0 | 0 | 0 | 0 | 0 | 2 | 0 |
| $\mu$-invariant(s) | - | - | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.