Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2+43883x+1595981\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z+43883xz^2+1595981z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+3554496x+1174133664\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4, 1331)$ | $1.2514253231558697972509456915$ | $\infty$ |
Integral points
\((4,\pm 1331)\)
Invariants
| Conductor: | $N$ | = | \( 100672 \) | = | $2^{6} \cdot 11^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-6528911929819136$ | = | $-1 \cdot 2^{14} \cdot 11^{9} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{321978368}{224939} \) | = | $2^{16} \cdot 11^{-3} \cdot 13^{-2} \cdot 17^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7231109415189586535815545807$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.28450840553349614610285468332$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9456942014037588$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.791914643149277$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.2514253231558697972509456915$ |
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| Real period: | $\Omega$ | ≈ | $0.26723193942462077551778718934$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.6753665292162070109755997326 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.675366529 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.267232 \cdot 1.251425 \cdot 8}{1^2} \\ & \approx 2.675366529\end{aligned}$$
Modular invariants
Modular form 100672.2.a.u
For more coefficients, see the Downloads section to the right.
| Modular degree: | 460800 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | additive | 1 | 6 | 14 | 0 |
| $11$ | $4$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $13$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 \( 264 = 2^{3} \cdot 3 \cdot 11 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 131 & 0 \\ 0 & 263 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 259 & 6 \\ 258 & 7 \end{array}\right),\left(\begin{array}{rr} 12 & 61 \\ 55 & 45 \end{array}\right),\left(\begin{array}{rr} 197 & 0 \\ 0 & 263 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 203 & 258 \\ 204 & 257 \end{array}\right),\left(\begin{array}{rr} 13 & 6 \\ 39 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[264])$ is a degree-$60825600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/264\Z)$.
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 | $2$ | \( 121 = 11^{2} \) |
| $11$ | additive | $72$ | \( 832 = 2^{6} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 7744 = 2^{6} \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 100672.u
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 572.a2, its twist by $-88$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-22}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.44.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.21296.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.2102060353536.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.2725888.4 | \(\Z/6\Z\) | 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 |
| $12$ | 12.0.7430465388544.6 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.21338037845105907031190198919760969728.2 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.9288285229884765407107758821952454656.1 | \(\Z/6\Z\) | 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 | ord | ord | ord | add | split | ss | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1 | 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 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.