Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-660508x+14521012\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-660508xz^2+14521012z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-53501175x+10425314250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(18, 1624)$ | $6.0084942402934792986492242176$ | $\infty$ |
| $(22, 0)$ | $0$ | $2$ |
| $(802, 0)$ | $0$ | $2$ |
Integral points
\( \left(-823, 0\right) \), \((18,\pm 1624)\), \( \left(22, 0\right) \), \( \left(802, 0\right) \), \((164797,\pm 66898650)\)
Invariants
| Conductor: | $N$ | = | \( 101400 \) | = | $2^{3} \cdot 3 \cdot 5^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $18353941222500000000$ | = | $2^{8} \cdot 3^{2} \cdot 5^{10} \cdot 13^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{1650587344}{950625} \) | = | $2^{4} \cdot 3^{-2} \cdot 5^{-4} \cdot 7^{3} \cdot 13^{-2} \cdot 67^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3859402698798733943001280348$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.16335148544124203397181676690$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9882017545674919$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.495243307835092$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.0084942402934792986492242176$ |
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| Real period: | $\Omega$ | ≈ | $0.18569490564411412919242321159$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2\cdot2\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.4629870840580033833058368033 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.462987084 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.185695 \cdot 6.008494 \cdot 64}{4^2} \\ & \approx 4.462987084\end{aligned}$$
Modular invariants
Modular form 101400.2.a.c
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3096576 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{1}^{*}$ | additive | 1 | 3 | 8 | 0 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
| $13$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 780 = 2^{2} \cdot 3 \cdot 5 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 391 & 160 \\ 470 & 321 \end{array}\right),\left(\begin{array}{rr} 623 & 0 \\ 0 & 779 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 471 & 470 \\ 50 & 311 \end{array}\right),\left(\begin{array}{rr} 479 & 310 \\ 0 & 779 \end{array}\right),\left(\begin{array}{rr} 777 & 4 \\ 776 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[780])$ is a degree-$1207664640$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/780\Z)$.
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 | $2$ | \( 4225 = 5^{2} \cdot 13^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 33800 = 2^{3} \cdot 5^{2} \cdot 13^{2} \) |
| $5$ | additive | $18$ | \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 101400.c
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1560.g2, its twist by $65$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{5}, \sqrt{-39})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{65})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | 16.0.137011437068313600000000.1 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | nonsplit | add | ord | ord | add | ord | ss | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 3 | - | 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 |
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.